aa r X i v : . [ m a t h . M G ] S e p NONLOCAL CURVATURE FLOWS
ANTONIN CHAMBOLLE, MASSIMILIANO MORINI, AND MARCELLO PONSIGLIONE
Abstract.
This paper aims at building a unified framework to deal witha wide class of local and nonlocal translation-invariant geometric flows.First, we introduce a class of generalized curvatures, and prove the exis-tence and uniqueness for the level set formulation of the correspondinggeometric flows.We then introduce a class of generalized perimeters, whose first varia-tion is an admissible generalized curvature. Within this class, we imple-ment a minimizing movements scheme and we prove that it approximatesthe viscosity solution of the corresponding level set PDE.We also describe several examples and applications. Besides recov-ering and presenting in a unified way existence, uniqueness, and ap-proximation results for several geometric motions already studied andscattered in the literature, the theory developed in this paper allows usto establish also new results.Keywords: Geometric evolution equations, Minimizing movements, Vis-cosity solutions.2000 Mathematics Subject Classification: 53C44, 49M25, 35D40.
Contents
1. Introduction 2
Part 1. Nonlocal Curvature Flows
82. Viscosity solutions: definition, properties, existence 82.1. Axioms of a nonlocal curvature 82.2. Viscosity solutions 92.3. Convergence of sets with uniform superjet 102.4. Semi continuous extensions of κ Part 2. Variational Nonlocal Curvature Flows
5. First variation of the perimeter 316. Examples of perimeters and their curvature 356.1. The Euclidean perimeter 366.2. The fractional mean curvature flow 366.3. General two body interaction perimeters 386.4. The flow generated by the regularized pre-Minkowski content 386.5. The shape flow generated by p -capacity 397. The minimizing movements approximation 437.1. The time-discrete scheme for bounded sets 437.2. The time discrete scheme for unbounded sets 467.3. The level-set approach 487.4. Convergence analysis 507.5. Perimeter descent 54References 571. Introduction
In this paper we present a unified approach to deal with a large class of possiblynonlocal geometric flows; i.e., evolutions of sets t E ( t ) governed by a law of theform(1.1) V ( x, t ) = − κ ( x, E ( t )) , where V ( x, t ) stands for the (outer) normal velocity of the boundary ∂E ( t ) at x andthe function κ ( · , E ) will be referred to as a generalized curvature of ∂E , in analogywith the classical theory.If the function κ depends only on how ∂E looks around x , then the flow is local innature. This is of course the case of the classical mean curvature flow , where κ ( · , E )is nothing but the mean curvature of ∂E , i.e. the first variation of the standardperimeter functional at E . On the other hand, for some of the relevant flows thathave been intensively studied in recent years the generalized curvature κ is trulynonlocal and depends on the global shape of the evolving set E ( t ) itself. It happensfor instance for fractional mean curvature flows , where the corresponding curvaturesare defined as the first variation of the so-called fractional perimeters , and hencerepresent the natural nonlocal counterparts of the classical mean curvature in thefractional framework ([4], [5], [22], [30]).As already made clear by the aforementioned examples, a relevant class of cur-vatures is given by those that can be seen as the first variation of some generalizedperimeters ; we refer to such a class as variational curvatures . It is important to ob-serve that when κ is variational, then (1.1) can be interpreted as the gradient flow ofthe corresponding perimeter, with respect to a suitable L Riemannian structure.In the case of the classical mean curvature flow, this observation underpins the minimizing movements algorithm implemented by Almgren-Taylor-Wang in theirpioneering work [1] (see also [25]).
ONLOCAL CURVATURE FLOWS 3
The strong formulation of the motion (1.1), which requires smoothness, faces thepossible formation of singularities in finite time. Thus, the evolution can only bedefined locally in time, which is clearly unsatisfactory from the applications pointof view. On the other hand, Brakke [3] proposed a weak formulation for motion bymean curvature that resulted in deep regularity results but had the disadvantageof producing a lack of uniqueness. These uniqueness issues are often overcome bythe more recent notion of generalized motion that is associated to the so-called levelset approach . Such an approach is based on representing the evolving set as thezero super-level set of a function u ( x, t ), which is defined for all times as viscositysolution to the (degenerate) parabolic partial differential equation(1.2) u t ( x, t ) + | Du ( x, t ) | κ ( x, { y : u ( y, t ) > u ( x, t ) } ) = 0 . The level-set method was proposed in [27], analytically validated in [18] for themotion by mean curvature and in [15] for more general local motions. In the caseof the classical mean curvature (and of several different local curvatures) viscositysolutions to (1.2) with a prescribed initial datum are unique. Note also that (1.2)prescribes that all the super-level sets of u evolve according to (1.1).The paper is divided into two parts: The main focus of Part 1 is to develop ageneral level set approach for the geometric motions (1.1), while Part 2 is aimed atimplementing a general minimizing movements scheme `a la Almgren-Taylor-Wangfor a large class of variational curvature motions, and at exploring the connectionsbetween the two approaches. In carrying out the program of Part 1 and Part 2,we recover and present in a unified way existence, uniqueness, and approximationresults for several geometric motions already studied and scattered in the literature,but we also establish new results (see the end of this Introduction).We now describe the content of the paper in more details. In Part 1 we introducethe class of generalized curvatures we deal with and then we set up the viscositytheory for the corresponding generalized level set equation (1.2).To be more specific, a generalized curvature κ is a function defined on the admis-sible pairs ( x, E ), where E is a C -set with compact boundary and x ∈ ∂E , andsuch that κ ( x, · ) is monotone non-increasing with respect to the inclusion betweensets touching at x and continuous with respect to C -convergence of sets. We alsoassume the translation invariance, i.e., κ ( x, E ) = κ ( x + y, E + y ) for all admissblepairs ( x, E ) and for all y ∈ R N .In order to fully exploit the second order viscosity solutions formalism, we needto extend the definition of the right-hand side of (1.2) to non-smooth sets. This isachieved by considering suitable lower and upper semi-continuous envelopes κ ∗ and κ ∗ of κ that are then employed to define viscosity subsolutions and supersolutions,respectively. The domain of definition of the relaxed curvatures κ ∗ and κ ∗ is madeup of the elements of the form ( x, p, X, E ), where E is now any measurable set, x ∈ ∂E , and ( p, X ) belongs to the second order super-jet (as far as κ ∗ is concerned)or sub-jet (as far as κ ∗ is concerned) of E at x . By construction, κ ∗ and κ ∗ turn outto be lower and upper semicontinuous, respectively, with respect to the Hausdorff As is often the case in viscosity solution approaches, we may in fact assume that the curvatureis a priori defined only for smoother sets, and will later on consider also stronger regularities.
A. CHAMBOLLE, M. MORINI, AND M. PONSIGLIONE convergence of sets and a suitable notion of uniform convergence of super- andsub-jets (see Definition 2.6).Remarkably, the above semicontinuity property is weaker than the semicontinuitywith respect to L -convergence, which is one of the main hypotheses in the approachof [28]. This significantly increases the class of admissible curvatures we can treat.For instance, the aforementioned fractional curvatures are not semicontinuous withrespect to the L -convergence and the corresponding L -relaxation would be useless(equal to −∞ for every closed set). Let us also notice that κ ∗ and κ ∗ are defined onlyon “geometrically meaningful” objects and this represents a further difference from[28], while the relaxation procedure used to define the semicontinuous envelopes of κ is reminiscent of the approach of Cardaliaguet and co-authors (see [6, 7, 8, 9, 10]).Section 2 is entirely devoted to setting up the viscosity formalism. The latter taskbeing accomplished, a general existence theorem for the level set formulation of(1.1) and for the class of generalized translation invariant curvatures we specifiedbefore is easily obtained through an application of the standard Perron method (seeTheorem 2.20).The drawback of such a generality is that the classical strategy to prove thecomparison principle and, in turn, the uniqueness of viscosity solutions may fail.Uniqueness is the main focus of Section 3, where we provide two different treatments,distinguishing between first order and second order geometric flows.Roughly speaking, we say that a geometric flow is of first order if the envelopes κ ∗ and κ ∗ do not depend on the second derivative variables (see condition (FO) atthe beginning of Subsection 3.1). Fractional mean curvature motions and the shapeflow generated by the p -capacity in R N are relevant examples of first order geometricmotions, see below. Uniqueness for such motions follows from the ComparisonPrinciple provided by Theorem 3.5. Let us mention that the main technical toolused in the first order uniqueness theory is represented by the well-known IlmanenInterposition Lemma (see [21, 6, 7]).The uniqueness theory for second order flows is harder and developed in Subsec-tion 3.2. In order to understand the source of difficulty, notice that semicontinuityproperties of κ ∗ and κ ∗ are not sufficient to conclude that the subsolution and thesupersolution inequalities extend to elements of the closure of the parabolic super-and sub-jets, respectively. This means that the usual machinery to establish unique-ness for second order equations, which is based on the celebrated Ishii’s Lemma (see[16]), cannot be applied. Indeed, Ishii’s Lemma states that if u is upper semicon-tinuous, v is lower semicontinuous, and u − v attains a (local) maximum at ( x, t ),then there exists at least one element belonging to both the closure of the parabolicsuper-jet of u and the closure of the parabolic sub-jet of v . Such a separating el-ement is obtained through a limiting procedure, which involves regularizations viainf- and sup-convolutions, the Alexandrov theorem on the a.e. second order dif-ferentiability of semi-convex functions, and a perturbation argument due to Jensen([16, Lemma A.3]). Since we lack the proper semicontinuity properties, we need toavoid “passing to the limit”. Our proof of the second order Comparison Principle(see Theorem 3.8) still uses all the aforementioned tools but combines them withsome new insight, allowing us to avoid the limiting procedure. The little price we ONLOCAL CURVATURE FLOWS 5 have to pay is a reinforced continuity assumption on κ (see beginning of Subsec-tion 3.2), which is nevertheless satisfied by all the relevant examples we have inmind.In Part 2 of the paper we take on a variational approach to geometric flows basedon the minimizing movements. To this aim, we introduce a class of functionalsreferred to as generalized perimeters . More precisely, denoting by M the class ofLebesgue-measurable sets, we call a generalized perimeter any translation invariant set function J : M → [0 , + ∞ ], which is insensitive to modifications on negligiblesets, finite on C -sets with compact boundary, lower semicontinuous with respectto L loc -convergence, and satisfying the following submodularity condition : For anymasurable sets E, F ⊂ R N ,(1.3) J ( E ∪ F ) + J ( E ∩ F ) ≤ J ( E ) + J ( F ) . It turns out that the latter condition is a convexity condition in the following sense:Extend J to L loc ( R N ) by enforcing the generalized coarea formula (4.3)(see [30]).Then, the submodularity property of J is equivalent to the convexity of the extendedfunctional (see [13]). A first important consequence of submodularity is that if J admits a first variation κ , then such a curvature is monotone. More in general, if J is smooth enough, then its first variation is an admissible generalized curvature.As mentioned before, the main achievement of Part 2 is the implementation of a generalized Almgren-Taylor-Wang minimizing movements scheme to approximategeometric motions associated with variational generalized curvatures. We recallthat, given an initial set E and a time step h >
0, such a scheme provides adiscrete-in-time evolution obtained by solving iteratively suitable incremental mini-mum problems. The energy to be minimized at each discrete time is the sum of thegeneralized perimeter and of a suitable dissipation that penalizes the L -distancefrom the boundary of the set obtained at the previous step. It turns out, once againdue to submodularity, that the discrete evolutions satisfy the comparison princi-ple. Adopting the point of view introduced in [12], we combine the minimizingmovements scheme with the level set framework. More precisely, given an initialfunction u , we let all its super-level sets evolve according to our discrete scheme,selecting at each time step the minimal solution (the maximal one would work aswell). In light of the discrete comparison principle, the evolving sets are themselvesthe super-levels sets of a discrete-in-time evolving function u h ( · , t ).We then study the limiting behavior of u h as the time step h ց consistency princi-ple : For all variational generalized curvatures, the discrete evolutions u h providedby the minimizing movements scheme converge (up to subsequences) to a viscositysolution of the level set equation. In particular, under the additional assumptionsthat guarantee uniqueness, the whole sequence converges (to the unique viscositysolution). Let us mention that in the case of the mean curvature motion the con-sistency between the level set and the minimizing movements approach has beenestablished in [12] (see also [17]). From the technical point of view, the convergenceanalysis combines several ingredients, among which we mention some careful esti-mates on the speed of propagation of the support of the initial function u and a A. CHAMBOLLE, M. MORINI, AND M. PONSIGLIONE subgradient inequality involving the generalized curvature, which is crucial in ob-taining the limiting sub(super)-solution property. The subgradient inequality is aconsequence of the analysis developed in Section 5 (see (5.3)). Note that since wedon’t have a regularity theory for the minimizers of the incremental problems, allour argument are necessarily variational in nature.A relevant consequence of our general consistency principle is that the generalizedperimeter of the super-level sets for which no fattening occurs during the evolutionis non-increasing in time (note that the no fattening condition is satisfied by almostall super-level sets). Moreover, we show that a suitable inner regularization ofthe generalized perimeter J , defined on open sets A as the inf − lim inf of J alongsequences of open sets approximating A from the interior (see Definition 7.20), isalways non-increasing in time (see Subsection 7.5).We conclude this introduction by highlighting some relevant examples and appli-cations of our general theory that are presented in the paper (see Section 6). Suchexamples are by no means exhaustive and serve indicating the scope of our theory:– Local motions:
The theory of local generalized mean curvature motions estab-lished in [15] fits into our theory as particular case.–
Fractional mean curvatures motions:
As mentioned before, these are the firstorder geometric flows associated with the so-called fractional perimeters. Existenceand uniqueness of viscosity solutions to the corresponding level set equation werealready established in [22] and are recovered here. On the other hand, the con-vergence of the minimizing movements scheme provided by our theory is new forthese motions and furnishes an approximation algorithm that is alternative to thethreshold-dynamics-based one studied in [5].–
Capacity flows:
Given 1 < p < N , consider the generalized perimeter that coin-cides with the p -capacity in R N on bounded sets of class C . It can be shown thatthe associated curvature κ ( x, E ) is given by | Dw E ( x ) | p , where w E denotes the ca-pacitary potential of E . Thus, the associated geometric motion is somewhat relatedto the Hele-Show type flows studied in [8, 9, 10] (see also [6, 7]). Our general resultsyield existence, uniqueness, and approximation via minimizing movements also forthis shape flow, which turns out to be of first order according to our terminolgy(see Subsection 6.5).– Second order nonlocal motions : Our theory includes all the generalized curva-tures treated in [28], that are in addition translation invariant. As already men-tioned, compared to our approach the theory of [28] requires stronger continuityassumptions on the Hamiltonians and more restrictive growth conditions that ruleout singular behavior of generalized curvatures along shrinking balls. As a furtherexample, which is not covered by the theory developed in [28] while fitting intoours, we mention here the generalized perimeter introduced in [2] in the frameworkof two-phase Image Segmentation. We will refer to it as regularized pre-Minkowskicontent of a set since it consists in a suitable regularization of the Lebesgue measureof the ρ -neighborhood of the essential boundary, for some fixed ρ >
0. The corre-sponding geometric motion was studied in [14]. In the latter work we computed theassociated generalized curvature and proved convergence of the minimizing move-ments scheme to a viscosity solution of the level set equation, but we were unable
ONLOCAL CURVATURE FLOWS 7 to establish uniqueness. The theory of the present paper allows us to recover theexistence and approximation results of [14] and, in addition, yields the uniquenessof the geometric motion.A final remark regarding the translation invariance and the continuity assump-tions on κ is in order. Concerning the former, we believe that it can be removedat the expense of some additional technical effort but within the main theoreticalframework introduced in this paper. On the other hand, the continuity assumptionsthat guarantee the stability property and the comparison principle are of a moresubtle and essential nature. They exclude from our theory some relevant irregularperimeters, as crystalline perimeters (that also would require a different specificviscosity level set formulation). It is not clear at the present which is (if any) theweakest continuity assumption on κ yielding the uniqueness of the geometric flow. A. CHAMBOLLE, M. MORINI, AND M. PONSIGLIONE
Part Nonlocal Curvature Flows
In this part we introduce the class of generalized curvatures we deal with. Then,we introduce the notion of viscosity solutions for the level set equation of the geo-metric flow, and we prove the existence and uniqueness of the solution.2.
Viscosity solutions: definition, properties, existence
We begin this section by introducing the class of generalized curvatures we willdeal with.2.1.
Axioms of a nonlocal curvature.
Let C be the class of subsets of R N , whichcan be obtained as the closure of an open set with compact C ℓ,β boundary, and let M be the class of all measurable subsets of R N . Throughout the paper ℓ ≥ β ∈ [0 ,
1] will be fixed, the reader may simply assume ℓ = 2, β = 0.In this part, we are given for every set E ∈ C a function κ ( x, E ) ∈ R definedat each point x ∈ ∂E , and referred to as the “curvature” of the set E at x . Thiscurvature will satisfy the following axioms:A) Monotonicity: If E, F ∈ C with E ⊆ F , and if x ∈ ∂F ∩ ∂E , then κ ( x, F ) ≤ κ ( x, E );B) Translational invariance: for any E ∈ C , x ∈ ∂E , y ∈ R N , κ ( x, E ) = κ ( x + y, E + y );C) Continuity: If E n → E in C and x n ∈ ∂E n → x ∈ ∂E , then κ ( x n , E n ) → κ ( x, E ).Axioms A), B) and C) are enough to prove the existence of a generalized solutionof the geometric flow 1.1. For axiom C), we say that E n → E in C if the boundariesconverge in the C ℓ,β sense (meaning for instance that they can be described locallywith a finite number of graphs of functions which converge in C ℓ,β ).By assumption C), for any ρ > c ( ρ ) := max x ∈ ∂B ρ max { κ ( x, B ρ ) , − κ ( x, R N \ B ρ ) } , (2.2) c ( ρ ) := min x ∈ ∂B ρ min { κ ( x, B ρ ) , − κ ( x, R N \ B ρ ) } , which are continuous functions of ρ >
0. Assumption D) below will guarantee thatthe curvature flow starting from a bounded set remains bounded at all times.D) Curvature of the balls: There exists
K > c ( ρ ) > − K > −∞ . Without assuming D) most of the results in this paper remain true, except thatthe flow starting from a given set with compact boundary will be defined possibly upto some time T ∗ < + ∞ where its boundary becomes unbounded and the frameworkof this paper cannot be applied anymore. The time T ∗ can be estimated from c ( ρ ).Observe that thanks to the monotonicity axiom A), the functions ρ c ( ρ ) and ρ c ( ρ ) are nonincreasing. ONLOCAL CURVATURE FLOWS 9
Viscosity solutions.
Here we introduce the level set formulation of the geo-metric evolution problem V = − κ , where V represents the normal velocity of theboundary of the evolving sets t E t , and we give a proper notion of viscositysolution. We refer to [20] for a general introduction of this approach for geomet-ric evolution problems. The level set approach consists in solving the followingparabolic Cauchy problem(2.4) ∂ t u ( x, t ) + | Du ( x, t ) | κ ( x, { y : u ( y, t ) ≥ u ( x, t ) } ) = 0 u (0 , · ) = u . Here and in the following, D and D stand for the spatial gradient and the spatialHessian matrix, respectively. Notice that if the superlevel sets of u are not smooth,the meaning of (2.4) is unclear. For this reason, it is necessary to use a definitionbased on appropriate smooth test functions whose curvature of level sets are welldefined. The appropriate setting is of course the framework of viscosity solutions.Let us first introduce a class of test functions appropriate for this problem.Following [23] (see also [14]), we introduce the family F of functions f ∈ C ∞ ([0 , ∞ )),such that f (0) = f ′ (0) = f ′′ (0) = 0, f ′′ ( r ) > r in a neighborhood of 0, f isconstant in [ M, + ∞ ) for some M >
0, and(2.5) lim ρ → + f ′ ( ρ ) c ( ρ ) = 0 , where c ( ρ ) is the function introduced in (2.1). We refer to [23, p. 229] for the proofthat the family F is not empty. Note that (2.5) (recall also (2.3)) implies(2.6) lim ρ → + f ′ ( ρ ) c ( f − ( ρ )) = 0 , since f − ( ρ ) > ρ for small values of ρ and c is decreasing.We fix T >
0, and look for geometric evolutions in the time interval [0 , T ]. In thefollowing, with a small abuse of language, we will say that a function g : R N × A → R ,with A ⊆ [0 , T ], is constant outside a compact set if there exists a compact set K ⊆ R N such that g ( · , t ) is constant in ( R N \ K ) for every t ∈ A (with the constantpossibly depending on t ). Definition 2.1.
Let ˆ z = (ˆ x, ˆ t ) ∈ R N × (0 , T ) and let A ⊆ (0 , T ) be any open intervalcontaining ˆ t . We will say that ϕ ∈ C ( R N × A ) is admissible at the point ˆ z = (ˆ x, ˆ t )if it is of class C in a neighborhood of ˆ z , if it is constant out of a compact set, and,in case Dϕ (ˆ z ) = 0, the following holds: there exists f ∈ F and ω ∈ C ∞ ([0 , ∞ ))with ω ′ (0) = 0, ω ( r ) > r = 0 such that | ϕ ( x, t ) − ϕ (ˆ z ) − ϕ t (ˆ z )( t − ˆ t ) | ≤ f ( | x − ˆ x | ) + ω ( | t − ˆ t | )for all ( x, t ) in R N × A . Definition 2.2.
An upper semicontinuous function u : R N × [0 , T ] → R (i.e. u ∈ U SC ( R N × [0 , T ])), constant outside a compact set, is a viscosity subsolution of theCauchy problem (2.4) if u (0 , · ) ≤ u , and for all z := ( x, t ) ∈ R N × (0 , T ) and forall C ∞ test functions ϕ such that ϕ is admissible at z and u − ϕ has a maximumat z (in the domain of definition of ϕ ) the following holds:i) If Dϕ ( z ) = 0, then ϕ t ( z ) ≤ ii) If the level set { ϕ ( · , t ) = ϕ ( z ) } is noncritical, then ϕ t ( z ) + | Dϕ ( z ) | κ ( x, { y : ϕ ( y, t ) ≥ ϕ ( z ) } ) ≤ . A lower semicontinuous function u (i.e. u ∈ LSC ( R N × [0 , T ])), constant outside acompact set, is a viscosity supersolution of the Cauchy problem (2.4) if u (0 , · ) ≥ u ,and for all z ∈ R N × (0 , T ) and for all C ∞ test functions ϕ such that ϕ is admissibleat z and u − ϕ has a minimum at z (in the domain of definition of ϕ ) the followingholds:i) If Dϕ ( z ) = 0, then ϕ t ( z ) ≥ { ϕ ( · , t ) = ϕ ( z ) } is noncritical, then ϕ t ( z ) + | Dϕ ( z ) | κ ( x, { y : ϕ ( y, t ) > ϕ ( z ) } ) ≥ . Finally, a function u is a viscosity solution of the Cauchy problem (2.4) if its uppersemicontinuous envelope is a subsolution and its lower semicontinuous envelope isa supersolution of (2.4).In this whole paper, we will use (with a small abuse of terminology) the termssubsolutions and supersolutions (omitting the locution “of the Cauchy problem(2.4)”) also for functions which do not satisfy the corresponding inequalities attime zero.While Definition 2.2 is quite natural, it has the drawback that the family ofpossible test functions is too restrictive to be handy. As usual in the viscositytheory, we will introduce suitable lower and upper semicontinuous extensions of κ , in particular to general measurable sets. This will allow to give definitionsequivalent to Definition 2.2, based on less smooth test functions as in Definition2.1 (See Definition 2.10), or on the notion of sub/superjets.2.3. Convergence of sets with uniform superjet.Definition 2.3.
Let E ⊆ R N , x ∈ ∂E , p ∈ R N , and X ∈ M N × N sym . We saythat ( p, X ) is in the superjet J , + E ( x ) of E at x if for every δ > U δ of x such that, for every x ∈ E ∩ U δ (2.7) ( x − x ) · p + 12 ( X + δI )( x − x ) · ( x − x ) ≥ . Moreover, we say that ( p, X ) is in the subjet J , − E ( x ) of E at x if ( − p, − X ) is inthe superjet J , + R N \ E ( x ) of R N \ E at x . Finally, we say that ( p, X ) is in the jet J E ( x ) of E at x if ( p, X ) ∈ J , + E ( x ) ∩ J , − E ( x ).The above definition of superjet of sets is consistent with the classical notion ofsuperjet of u.s.c. functions. Indeed, if u is an u.s.c. function, then ( p, X ) is inthe superjet of u at x if and only if ( p, X ) is in the superjet of the superlevel set { u ≥ u ( x ) } , according to Definition 2.3. In the same way, the above definition ofsubjet is consistent with the classical notion of subjet of l.s.c. functions. Remark 2.4.
It can be checked that the condition ( p, X ) ∈ J , + E ( x ) is equiv-alent to ( λp, λX + µ p ⊗ p ) ∈ J , + E ( x ) for all λ > µ ∈ R . Thus,( p, X ) ∈ J , + E ( x ) if and only if ( p | p | , | p | π p ⊥ Xπ p ⊥ ) ∈ J , + E ( x ), where π p ⊥ denotesthe projection operator on p ⊥ := { v ∈ R N : p · v = 0 } . Note that if E = { u ≥ u ( x ) } , ONLOCAL CURVATURE FLOWS 11 with u of class C and Du = 0 on ∂E , then, setting p := Du ( x ) and X := D u ( x ), p | p | is the inner normal to { u ≥ u ( x ) } at x , while | p | π p ⊥ Xπ p ⊥ represents the thesecond fundamental form of ∂ { u ≥ u ( x ) } (oriented with the external normal) at x .Let us introduce the notion of uniform superjet. Definition 2.5.
Let E n ⊆ R N and x ∈ ∂E n . We say that the ( p n , X n )’s are in thesuperjet J , + E n ( x ) uniformly, if for every positive δ > U δ of x (independent of n ) such that, for all n ∈ N ,(2.8) ( x − x ) · p n + 12 ( X n + δI )( x − x ) · ( x − x ) ≥ x ∈ E n ∩ U δ . In the following, given a set E ⊂ R N , E c := R N \ E denotes its complement.We also recall that a sequence of closed sets C n converges to a closed set C in theHausdorff metric ( C n H → C ) ifmax (cid:26) sup x ∈ C n dist( x, C ) , sup x ∈ C dist( x, C n ) (cid:27) → n → ∞ . Definition 2.6.
We say that ( p n , X n , E n ) converge to ( p, X, E ) with uniform su-perjet at x if E n → E in the Hausdorff sense, the ( p n , X n )’s are in the superjet J , + E n ( x ) uniformly and ( p n , X n ) → ( p, X ) as n → ∞ .Moreover, we say that ( p n , X n , E n ) converge to ( p, X, E ) with uniform subjet at x if ( − p n , − X n , E cn ) converge to ( − p, − X, E c ) with uniform superjet.2.4. Semi continuous extensions of κ . We now introduce two suitable lowerand upper semicontinuous extensions of κ , which will be instrumental in developingthe level set formulation of the geometric flow. This is reminiscent of the approachin [6, 7] for evolution of “tubes” by geometric motions (see also [8, 9, 10]). For every F ⊆ R N with compact boundary and ( p, X ) ∈ J , + F ( x ), we define(2.9) κ ∗ ( x, p, X, F ) := sup n κ ( x, E ) : E ∈ C , E ⊇ F , ( p, X ) ∈ J , − E ( x ) o Analogously, for any ( p, X ) ∈ J , − F ( x ) we set(2.10) κ ∗ ( x, p, X, F ) = inf n κ ( x, E ) : E ∈ C , ˚ E ⊆ F , ( p, X ) ∈ J , + E ( x ) o . (It is clear that κ ∗ only depends on the closure of F while κ ∗ depends on its interior,in practice the first one will be evaluated at superlevels of usc functions, while thesecond one at strict superlevels of lsc functions.)Clearly, it follows from the monotonicity property A) that if E ∈ C , and ( p, X ) ∈J E ( x ), then κ ∗ ( x, p, X, E ) = κ ∗ ( x, p, X, E ) = κ ( x, E ). Notice that the monotonicityof κ clearly extends to κ ∗ and κ ∗ . More precisely, κ ∗ ( x, p, X, E ) ≥ κ ∗ ( x, p, X, F )(resp. κ ∗ ( x, p, X, E ) ≥ κ ∗ ( x, p, X, F )) whenever E ⊆ F and ( p, X ) ∈ J , + E ( x ) ∩J , + F ( x ) (resp. ( p, X ) ∈ J , − E ( x ) ∩ J , − F ( x )).In the next Lemma we show that κ ∗ and κ ∗ are the l.s.c. and the u.s.c envelopeof κ with respect to the convergence defined in Definition 2.6, respectively. Lemma 2.7.
Let F ⊆ R N with compact boundary. Then, κ ∗ ( x, p, X, F ) = inf lim inf n κ ( x, E n ) where the infimum is over all ( p n , X n , E n ) → ( p, X, F ) with uniform superjet at x ; κ ∗ ( x, p, X, F ) = sup lim sup n κ ( x, E n ) where the supremum is over all ( p n , X n , E n ) → ( p, X, F ) with uniform subjet at x .Proof. We will prove the statement only for κ ∗ , the other case being analogous. Westart by showing that there exists a sequence ( p, X n , E n ) such that ( p, X n , E n ) → ( p, X, F ) with uniform superjet at x and κ ∗ ( x, p, X, F ) = lim n κ ( x, E n ). To this aim,recall that by definition of κ ∗ ( x, p, X, F ), for every n ∈ N there exists a set ˜ E n ∈ C with F ⊆ ˜ E n , ( p, X ) ∈ J , − ˜ E n ( x ) and such that 0 ≤ κ ∗ ( x, p, X, F ) − κ ( x, ˜ E n ) ≤ n .By the monotonicity of κ we may also assume that ˜ E n → F in the Hausdorff metric.Moreover, by the continuity assumption C) we can suitably modify the sequence˜ E n so that, in addition to the previous properties, we have ( p, X + δ n I ) ∈ J , − ˜ E n ( x ), ∂ ˜ E ε ∩ ∂F = { x } and 0 ≤ κ ∗ ( x, p, X, F ) − κ ( x, ˜ E n ) ≤ n for some suitable δ n ց E n according to the following inductive procedure.Assume that E , . . . , E n have been defined with the following properties:1) F ⊆ E n ⊆ E n − ⊆ . . . ⊆ E ,2) E i ⊆ ˜ E i for i = 1 , . . . n ,3) ∂E i ∩ ∂F = { x } for every i = 1 , . . . n
4) ( p, X + δ i I ) ∈ J E i ( x ) for every i = 1 , . . . n .Since X + δ n +1 I < X + δ n I, X + δ n +1 I < X + δ n +1 I, recalling that ( p, X + δ n I ) ∈ J E n ( x ) and ( p, X + δ n +1 I ) ∈ J , − ˜ E n +1 ( x ) we can easilyconstruct E n +1 ∈ C such that( p, X + δ n +1 I ) ∈ J E n +1 ( x ) , F ⊆ E n +1 ⊆ ˜ E n +1 , E n +1 ⊆ E n , ∂E n +1 ∩ ∂F = { x } . The sequence E n just constructed still converges to F in the sense of Hausdorff.Moreover, since E n is monotone decreasing, we have ( p, X + δ n I, E n ) → ( p, X, F )with uniform superjet. Note also that since ( p, X ) ∈ J , − E n ( x ), by (2.9) we imme-diately have κ ∗ ( x, p, X, F ) ≥ κ ( x, E n ) for all n . Finally, by the monotonicity of κ , κ ∗ ( x, p, X, F ) ≥ lim sup n κ ( x, E n ) ≥ lim inf n κ ( x, E n ) ≥ lim n κ ( x, ˜ E n ) = κ ∗ ( x, p, X, F ) . Now, let ( p n , X n , F n ) → ( p, X, F ) with uniform superjet at x . Let j ∈ N and let E j the set constructed above. Notice that F n is definitively contained in x + R n ( E j − x ),where R n is any rotation such that R n ( p ) = p n . By the monotonicity assumptionA) and the continuity assumption C) on κ we deduce(2.11) κ ( x, E j ) = lim n κ ( x, x + R n ( E j − x )) ≤ lim inf n κ ( x, F n ) . We conclude that lim inf n κ ( x, F n ) ≥ lim j κ ( x, E j ) = κ ∗ ( x, p, X, F ) . (cid:3) ONLOCAL CURVATURE FLOWS 13
Lemma 2.8.
Let ϕ n , ϕ ∈ C ( R N ) be constant outside a compact set K (independentof n ). Assume that ϕ n → ϕ uniformly and ϕ n → ϕ in C ( B ( x, δ )) for some δ > and x ∈ R N with Dϕ ( x ) = 0 . If x n → x , then (2.12) κ ∗ ( x, Dϕ ( x ) , D ϕ ( x ) , { ϕ ≥ ϕ ( x ) } ) ≤ lim inf n κ ∗ ( x n , Dϕ n ( x n ) , D ϕ n ( x n ) , { ϕ n ≥ ϕ n ( x n ) } ) and (2.13) κ ∗ ( x, Dϕ ( x ) , D ϕ ( x ) , { ϕ > ϕ ( x ) } ) ≥ lim sup n κ ∗ ( x n , Dϕ n ( x n ) , D ϕ n ( x n ) , { ϕ n > ϕ n ( x n ) } ) . Proof.
Up to a subsequence, we can assume that K n := x − x n + { ϕ n ≥ ϕ n ( x n ) } converge in the sense of Hausdorff to some closed set ˜ K contained in K := { ϕ ≥ ϕ ( x ) } . Since Dϕ ( x ) = 0, then ( Dϕ n ( x n ) , D ϕ n ( x n ) , K n ) → ( Dϕ ( x ) , D ϕ ( x ) , ˜ K )with uniform superjet at x . Thus, by Lemma 2.7 and the monotonicity of κ ∗ wemay conclude κ ∗ ( x, Dϕ ( x ) , D ϕ ( x ) , K ) ≤ κ ∗ ( Dϕ ( x ) , D ϕ ( x ) , ˜ K ) ≤ lim inf n κ ∗ ( x, Dϕ n ( x n ) , D ϕ n ( x n ) , K n ) , which proves (2.12). The proof of (2.13) is identical. (cid:3) Remark 2.9.
In [28], a class of nonlocal Hamiltonians H ( x, p, X, F ) that are lowersemicontinuous with respect to the L convergence of sets and the standard conver-gence of the other variables is considered. Notice that such Hamiltonians are alsolower semicontinuous with respect to the convergence with uniform superjets intro-duced in Definition 2.6. Indeed, if ( p n , X n , F n ) → ( p, X, F ) with uniform superjetat x , then one can show that F n ∪ F → F in L loc . Thus, H ( x, p, X, F ) ≤ lim inf n H ( x, p n , X n , F n ∪ F ) ≤ lim inf n H ( x, p n , X n , F n ) , where the last inequality follows from the monotonicity assumption on H withrespect to the set variable.2.5. Equivalent definition of the viscosity solutions.
We now can give a sec-ond definition of viscosity solutions of (2.4). It is seemingly more restrictive thanthe previous Definition 2.2, but we will check later on that it is equivalent.
Definition 2.10.
An upper semicontinuous function u : R N × [0 , T ] → R , constantoutside a compact set, is a viscosity subsolution of the Cauchy problem (2.4) if u (0 , · ) ≤ u , and for all z := ( x, t ) ∈ R N × (0 , T ) and all ϕ admissible at z , suchthat u − ϕ has a maximum at z (in the domain of definition of ϕ ) we havei) If Dϕ ( z ) = 0, then ϕ t ( z ) ≤ Dϕ ( z ) = 0, then(2.14) ϕ t ( z ) + | Dϕ ( z ) | κ ∗ (cid:0) x, Dϕ ( z ) , D ϕ ( z ) , { y : ϕ ( y, t ) ≥ ϕ ( z ) } (cid:1) ≤ . The definition of viscosity supersolutions and of viscosity solutions are given ac-cordingly in the obvious way.
We will need the following technical lemma.
Lemma 2.11.
Let Q ∈ C ∞ ( R N ) , Q ≥ with equality only for x = 0 , Q convex in B , and constant in R N \ B .Let ϕ ∈ C ( R N ) , and let ¯ x be such that Dϕ (¯ x ) = 0 . For every η ∈ R set ϕ η ( x ) := ϕ ( x ) + ηQ ( x − ¯ x ) . Then, for almost every η small enough the ϕ (¯ x ) -level set of ϕ η is not critical.Proof. Let B (¯ x, δ ) be a neighborhood of ¯ x where Dϕ = 0. Clearly in B (¯ x, δ ) Dϕ + ηDQ ( x − ¯ x ) = 0 if η is small enough.Then, in R N \ B (¯ x, δ/ C ∞ function x
7→ − ϕ ( x ) − ϕ (¯ x ) Q ( x − ¯ x ) , and by Sard’s theorem, we know that for a.e. η >
0, the level set η of this functionis not critical. This means that for all x B (¯ x, δ/
2) with ϕ ( x ) + ηQ ( x − ¯ x ) = ϕ (¯ x ),one has0 = 1 Q ( x − ¯ x ) ( − Dϕ ( x ) Q ( x − ¯ x ) + DQ ( x − ¯ x )( ϕ ( x ) − ϕ (¯ x )))= − Q ( x − ¯ x ) ( Dϕ ( x ) + ηDQ ( x − ¯ x )) , so that the ϕ (¯ x )-level set of ϕ η is not critical. (cid:3) Remark 2.12.
As it is standard in the theory of viscosity solutions, the maximumin Definition 2.10 of subsolutions can be assumed to be strict (and similarly forsupersolutions). Assume for instance that u is a subsolution, u − ϕ has a maximumat some (¯ x, ¯ t ), with ϕ admissible at (¯ x, ¯ t ). If Dϕ (¯ x, ¯ t ) = 0 we replace ϕ by ϕ s ( x, t ) := ϕ ( x, t ) + s min { , | x − ¯ x | + | t − ¯ t | } . Then the maximum of u − ϕ s at (¯ x, ¯ t ) is strict, and we recover the inequality (2.14)for ϕ by letting s → κ ∗ provided byLemma 2.8. If Dϕ (¯ x, ¯ t ) = 0, we choose f ∈ F as in Definition 2.1, and replace ϕ by ˜ ϕ ( x, t ) := ϕ ( x, t ) + f ( | y − x | ) + | t − s | . We still have D ˜ ϕ (¯ x, ¯ t ) = 0, ˜ ϕ is admissible at (¯ x, ¯ t ), ˜ ϕ t (¯ x, ¯ t ) = ϕ t (¯ x, ¯ t ) and now themaximum of u − ˜ ϕ is strict.Moreover, one can assume without loss of generality that ϕ is smooth. If Dϕ (¯ x, ¯ t ) = 0, this follows again by Lemma 2.8 and by standard mollification argu-ments. If Dϕ (¯ x, ¯ t ) = 0, since ϕ is admissible at z , there are f ∈ F and ω ∈ C ∞ ( R )with ω ′ (0) = 0 such that | ϕ ( x, t ) − ϕ (¯ x, ¯ t ) − ϕ t ( x, t )( t − ¯ t ) | ≤ f ( | x − ¯ x | ) + ω ( t − ¯ t ) . Then it is enough to replace ϕ by ψ ( x, t ) := ϕ t ( z )( t − ¯ t ) + f ( | x − ¯ x | ) + ω ( t − ¯ t ) . Finally, in view of Lemma 2.11 one can assume that the superlevel set of ϕ inDefinition 2.10, ii) is not critical. ONLOCAL CURVATURE FLOWS 15
Remark 2.13.
Note that in view of the above observations we have shown, inparticular, that Definitions 2.2 and 2.10 are equivalent.We now introduce the notion of parabolic sub/superjets.
Definition 2.14.
Let u : R N × (0 , T ) → R be upper semicontinuous at ( x, t ). Wesay that ( a, p, X ) ∈ R × R N × M N × Nsym is in the parabolic superjet P , + u ( x, t ) of u at ( x, t ), if u ( y, s ) ≤ u ( x, t ) + a ( s − t ) + p · ( y − x ) + 12 ( X ( y − x )) · ( y − x ) + o ( | t − s | + | x − y | )for ( y, s ) in a neighborhood of ( x, t ). If u is lower semicontinuous at ( x, t ) we can de-fine the parabolic subjet P , − u ( x, t ) of u at ( x, t ) as P , − u ( x, t ) := −P , + ( − u )( x, t ).The next lemma provides another equivalent definition of viscosity solutions interms of the superlevel sets of u and the corresponding parabolic jets. Lemma 2.15.
Let u be a viscosity subsolution of (2.4) in the sense of Defini-tion 2.10. Then, for all ( x, t ) in R N × (0 , T ) , if ( a, p, X ) ∈ P , + u ( x, t ) , and p = 0 ,then (2.15) a + | p | κ ∗ ( x, p, X, { y : u ( y, t ) ≥ u ( x, t ) } ) ≤ . A similar statement holds for supersolutions.Proof of Lemma 2.15.
By definition of parabolic subjets, given ε > δ >
0, thereexists a neighborhood U of ( x, t ) in R N × (0 , t ] where u ( y, s ) ≤ u ( x, t ) + ( a − ε )( s − t ) + p · ( y − x ) + 12 ( X + δI )( y − x ) · ( y − x )with a strict inequality if y = x or s < t . Let p ε,δ : R N × [0 , T ] be a continuousfunction such that p ε,δ ( y, s ) = u ( x, t ) + ( a − ε )( s − t ) + p · ( y − x ) + 12 ( X + δI )( y − x )) · ( y − x )in U , p ε,δ ≥ u in R N × (0 , t ], with equality only on ( x, t ), p ε,δ ≥ u + c in ( R N × (0 , t ]) \ U for some c >
0, and p ε,δ is constant (possibly depending on time) in ( R N \ K ), wherealso u is constant. Consider a decreasing sequence ψ k of smooth functions, suchthat ψ k is constant in ( R N \ K ), inf k ψ k = u , and ψ k ≥ u + 1 /k . Such a sequenceexists because u is upper-semicontinuous. Let ϕ k := min { ψ k , p ε,δ } , so that ϕ k > u in R N × (0 , t ], except at ( x, t ) where equality holds, and ϕ k = p ε,δ near ( x, t ).For any n ∈ N large enough, the function ( y, s ) u ( y, s ) − ϕ k ( y, s ) − / [ n ( t − s )]attains a maximum at a point z n = ( y n , s n ) ∈ (0 , t ) × R N , where z n → z = ( x, t ) as n → ∞ . Moreover, Dϕ k ( z n ) = p + ( X + δI )( y n − x ) = 0 for n large.Hence, by Definition 2.10 of a viscosity subsolution, ϕ kt ( z n ) + 1 n ( t − s n ) + | Dϕ k ( z n ) | κ ∗ ( y n , Dϕ k ( z n ) , D ϕ k ( z n ) , { ϕ k ( · , s n ) ≥ ϕ k ( z n ) } ) ≤ . Since ∂ t ϕ kt ( z n ) = a − ε it follows that a + | Dϕ k ( z n ) | κ ∗ ( y n , Dϕ k ( y n , s n ) , D ϕ k ( y n , s n ) , { ϕ k ( · , s n ) ≥ ϕ k ( y n , s n ) } ) ≤ ε. Letting n → ∞ and invoking Lemma 2.8 we obtain a + | Dϕ k ( x, t ) | κ ∗ ( x, Dϕ k ( x, t ) , D ϕ k ( x, t ) , { ϕ k ( · , t ) ≥ ϕ k ( x, t ) } ) ≤ ε , that is a + | p | κ ∗ ( x, p, X + δI, { ϕ k ( · , t ) ≥ ϕ k ( x, t ) } ) ≤ ε. Now, as { ϕ k ( · , t ) ≥ ϕ k ( x, t ) } is a decreasing sequence converging to { u ( · , t ) ≥ u ( x, t ) } ), we get that ( p, X + δI, { ϕ k ( · , t ) ≥ ϕ k ( x, t ) } ) → ( p, X + δI, { u ( · , t ) ≥ u ( x, t ) } ) with uniform superjet, as k → ∞ . Therefore, by Lemma 2.7 we infer a + | p | κ ∗ ( x, p, X + δI, { u ( · , t ) ≥ u ( x, t ) } ) ≤ ε. The conclusion follows by applying again Lemma 2.7, after observing that( p, X + δI, { u ( · , t ) ≥ u ( x, t ) } ) → ( p, X, { u ( · , t ) ≥ u ( x, t ) } )with uniform superjet, as δ → (cid:3) In the next lemma we show that equation (2.4) is satisfied in a suitable viscositysense also for t = T . To this purpose, we notice that the notion of admissible testfunctions ϕ given in Definition 2.1 can be extended also at points (ˆ x, T ) ∈ R N × { T } without any change. This is a classical fact, we adapt here the proof in [23]. Lemma 2.16.
Let u ∈ U SC ( R N × [0 , T ]) be a subsolution of (2.4) . If ϕ is ad-missible at (ˆ x, T ) and u − ϕ has a (one-sided w.r.t. time) maximum in R N × [0 , T ] at (ˆ x, T ) , then i) and ii) of Definition 2.10 are satisfied at (ˆ x, T ) . An analogousstatement holds for supersolutions.Proof. First assume that Dϕ (ˆ x, T ) = 0. As usual, we can assume that the maximumis strict. For n large enough, the function u ( x, t ) − ϕ ( x, t ) − / [ n ( T − t )] has amaximum at a point z n := (ˆ x n , t n ) ∈ R N × (0 , T ) converging to z := (ˆ x, T ) as n → ∞ . Since u is a subsolution in R N × (0 , T ), for n large enough we have ϕ t ( z n ) + 1 n ( T − t n ) + | Dϕ ( z n ) | κ ∗ (cid:0) x n , Dϕ ( z n ) , D ϕ ( z n ) , { ϕ ( · , t n ) ≥ ϕ ( z n ) } (cid:1) ≤ . Letting n → ∞ , the conclusion follows from Lemma 2.8.If now Dϕ ( z ) = 0, we follow the lines of [23, Proposition 1.3]. Since ϕ is admis-sible at z , there are f ∈ F and ω ∈ C ∞ ( R ) with ω ′ (0) = 0 such that | ϕ ( x, t ) − ϕ ( z ) − ϕ t ( z )( t − T ) | ≤ f ( | x − ˆ x | ) + ω ( t − T ) . Set ψ ( x, t ) = ϕ t ( z )( t − T ) + 2 f ( | x − ˆ x | ) + 2 ω ( t − T ) ,ψ n ( x, t ) = ψ ( x, t ) − n ( T − t ) . (2.16)We have that u − ψ has a strict maximum at z . Hence for n large enough u − ψ n has a maximum at z n = (ˆ x n , t n ) ∈ R N × (0 , T ), with z n → z . As ψ n is admissibleat z n and u is a subsolution, we have(2.17) ϕ t ( z ) + ω ′ ( t n − T )+ 2 f ′ ( | ˆ x n − x | ) κ ∗ (ˆ x n , Dψ n (ˆ x n ) , D ψ n (ˆ x n ) , { ψ n ( · , t n ) ≥ ψ n ( z n ) } ) ≤ ONLOCAL CURVATURE FLOWS 17 if ˆ x n = x , while ϕ t ( z ) + ω ′ ( t n − T ) ≤ x n = x . Note that { ψ n ( · , t n ) ≥ ψ n ( z n ) } = R N \ B f − ( | ˆ x n − x | ) ( x ). Letting n → ∞ , we get ϕ t ( z ) ≤ u is a subsolution. (cid:3) Remark 2.17.
A similar reasoning also shows that the alternative characterizationof sub- and super-solutions provided by Lemma 2.15 holds also at points of the form( x, T ).2.6.
Existence of a viscosity solution.
Let u : R N R be a continuous func-tion, constant out of a compact set K . The existence of a viscosity solution to theCauchy problem (2.4) follows by standard arguments once the existence of at leastone supersolution and a stability property for supersolutions are established. Tothis purpose, we first prove a confinement condition. Lemma 2.18.
Let
R, T > be fixed. There exists a constant R ′ > R such that if u ∈ U SC ( R N × [0 , T ]) is a subsolution of (2.4) with u ( x, ≤ C for | x | ≥ R , then u ( x, t ) ≤ C for | x | > R ′ and t ∈ [0 , T ] . Proof.
Let ψ ∈ C ∞ ([0 , + ∞ )) be such that ψ ′ (0) = 0, ψ ( s ) ≡ C for s ≥ R , ψ strictly decreasing in [0 , R ], and ψ ( | x | ) ≥ u ( x,
0) for all x ∈ R N . We now constructa test function ϕ , by letting all the superlevel sets of ψ ( | · | ) expand with constantnormal velocity equal to K , where K is the constant appearing in (2.3). Precisely,we set(2.18) ϕ ( x, t ) := ψ ( | x | − Kt ) if | x | ≤ R + Kt , C otherwise.The lemma is proven (with R ′ := 2 R + KT ) once we show that ϕ ≥ u . Assumetoward a contradiction that there exists δ > x, t ) ∈ R N × (0 , T ) such that( u − ϕ )( x, t ) > δt . Then, setting ϕ δ ( x, t ) := ϕ ( x, t ) + δt , we have max R N × [0 ,T ] u − ϕ δ >
0. Let z δ := ( x δ , t δ ) be a maximum point and note that necessarily t δ > Dϕ δ ( z δ ) = 0, recalling the definition of a subsolution we get the contradiction0 ≤ δ = ( ϕ δ ) t ( z δ ) ≤
0. If Dϕ δ ( z δ ) = 0, recalling again Definition 2.10 of subsolutionwe get the contradiction δ = ϕ t ( z δ ) + δ − K | Dϕ ( z δ ) | < ϕ t ( z δ ) + δ + | Dϕ ( z δ ) | κ ∗ (cid:0) x δ , Dϕ ( z δ ) , D ϕ ( z δ ) , { ϕ ( · , t δ ) ≥ ϕ ( z δ ) } (cid:1) ≤ . (cid:3) It is very easy to show that, as in the classical case, the maximum of two subso-lutions is still a subsolution. In the following we show that the notion of subsolutionis stable also with respect to taking upper relaxed limits.
Proposition 2.19.
Let ( u n ) n ≥ be a sequence of viscosity subsolutions such that u n = c n in ( R N \ K ) × [0 , T ) , for some constant c n ∈ R and some compact K ⊆ R N .Let, for any z = ( x, t ) , (2.19) u ∗ ( z ) = lim r ↓ sup (cid:26) u n ( ζ ) : | z − ζ | ≤ r , n ≥ r (cid:27) . If u ∗ ( z ) < + ∞ for all z , then u ∗ is a subsolution. Of course, a symmetric result holds for supersolutions.
Proof.
Let A ⊂ (0 , T ) be an open interval and let ϕ : R N × A → R be an admissibletest function at z = ( x, t ) with ϕ ( · , s ) = C ( s ) in ( R N \ ˜ K ) × A for some compactset ˜ K and for all s ∈ A , and such that u − ϕ has a strict maximum at z . Assumefirst that Dϕ ( z ) = 0. Let z n be a maximum point of u n − ϕ in ( K ∪ ˜ K ) × ¯ A . Bystandard arguments it follows that z n → z . Since for every nϕ t ( z n ) + | Dϕ ( z n ) | κ ∗ (cid:0) x, Dϕ ( z n ) , D ϕ ( z n ) , { y : ϕ ( y, t ) ≥ ϕ ( z n ) } (cid:1) ≤ , by Lemma 2.8 we conclude that ϕ t ( z ) + | Dϕ ( z ) | κ ∗ (cid:0) x, Dϕ ( z ) , D ϕ ( z ) , { y : ϕ ( y, t ) ≥ ϕ ( z ) } (cid:1) ≤ . If now Dϕ ( z ) = 0, we follow the lines of [23, Proposition 1.3]. Since ϕ is admis-sible at z , there are δ > f ∈ F and ω ∈ C ∞ ( R ) with ω ′ (0) = 0, ω ( t ) > t > | ϕ ( y, s ) − ϕ ( z ) − ϕ t ( z )( s − t ) | ≤ f ( | x − y | ) + ω ( s − t )for all ( y, s ) ∈ R N × A . Set(2.20) ψ ( y, s ) = ϕ t ( z )( s − t ) + 2 f ( | y − x | ) + 2 ω ( s − t ) . Note that u − ψ has a unique maximum at z in R N × A . Let z n = ( y n , s n ) be amaximum point of u n − ψ in R N × ¯ A . Then z n → z . If Dψ ( z n ) = 0, then y n = x and ψ n is admissible at z n thanks to (2.20). We deduce ϕ t ( z ) + ω ′ ( s n − t ) ≤ y n = x , ψ is still admissible at z n and we have(2.21) ϕ t ( z ) + ω ′ ( s n − t )+ 2 f ′ ( | y n − x | ) κ ∗ ( y n , Dψ n ( y n ) , D ψ ( y n ) , { ψ ( · , s n ) ≥ ψ ( z n ) } ) ≤ . Note that { ψ ( · , s n ) ≥ ψ ( z n ) } = R N \ B f − ( | y n − x | ) ( x ). Letting n → ∞ , we get ϕ t ( z ) ≤ u is a subsolution. (cid:3) We now can state a general existence result:
Theorem 2.20.
Let u : R N → R be a uniformly continuous function with u = C for | x | ≥ R . Let R ′ be the constant given by Lemma 2.18. Then, there exists aviscosity solution u : R N × [0 , T ] → R of (2.4) with u = C for | x | ≥ R ′ .Proof. We briefly sketch the proof of this result which is very classical, see [16, 23],and based on Perron’s method. Let ϕ be the function defined in (2.18), and noticethat it is a supersolution. Then, we can set u ( x, t ) = inf { u ( x, t ) : u is a supersolution of the Cauchy problem (2.4) } . The fact that u is bounded from below follows easily by using a smooth barrier asin Lemma 2.18. For instance, the barrier C + λ ( C − ϕ ) will work for λ so largethat C + λ ( C − ψ ( | · | )) ≤ u ( · , u = C for | x | ≥ R ′ (thisis also is a consequence of the fact that u ( · ,
0) = u , which we explain at the end ofthis proof). Let u ∗ , u ∗ be the upper and lower semicontinuous envelope of u . Thefact that u ∗ is a supersolution follows from Proposition 2.19, observing that at each ONLOCAL CURVATURE FLOWS 19 point ( x, t ) we can find a suitable sequence of supersolutions ( u n ) n ≥ , constant on( R N \ B R ′ ) × [0 , T ) (see Lemma 2.18), whose relaxed lower limit is u ∗ ( x, t ).The fact that u ∗ is a subsolution is standard and obtained by contradiction,assuming that at some point ¯ z = (¯ x, ¯ t ) of (strict) contact with a test function ϕ ≥ u ∗ , ϕ does not satisfy (2.14). If Dϕ (¯ z ) = 0, one can use the test function ϕ toconstruct a new supersolution u < u ∗ in a neighborhood of ¯ z , thus contradicting theminimality of ¯ u . To treat the case Dϕ (¯ z ) = 0 one repeats the same construction,but (as in the proof of [23, Prop. 1.3]) with ϕ replaced by ψ ( x, t ) = ϕ (¯ z ) + ϕ t (¯ z )( t − ¯ t ) + 2 f ( | x − ¯ x | ) + 2 ω ( t − ¯ t ) . Finally, the initial condition u ( · ,
0) = u can be shown as in the last part of theproof of [23, Theorem 1.8]. (cid:3) Uniqueness of viscosity solutions
In this section we will prove that, under some additional assumptions, (2.4)admits a unique viscosity solution. In the first subsection, we consider first order geometric flows, corresponding to the case where the relaxed curvatures κ ∗ and κ ∗ depend only on the first order super-jet and sub-jet, respectively. Examples ofrelevant first-order flows are given in Section 6.In the second subsection, we deal with truly second order flows, under an addi-tional uniform continuity assumption on the nonlocal curvature κ .Before entering the details of the uniqueness theory, it is convenient to give thefollowing definition and state an auxiliary lemma. Definition 3.1.
We say that a set valued function F : [0 , T ] → M is a subsolutionof the geometric flow (1.1) if χ F ( t ) is a viscosity subsolution of (2.4) in the senseof Definition 2.10. The definition of supersolutions and solutions of the geometricflow are analogous. Lemma 3.2.
Let u be a subsolution of (2.4) . Then, for every s ∈ R the set function t → F ( t ) := { u ( · , t ) ≥ s } is a subsolution of the geometric flow (1.1) , according withDefinition 3.1. The analogous statement holds for supersolutions. The proof is classical and follows by approximating the Heavyside function as asupremum of smooth increasing functions H n , so that χ F ( x, t ) = sup n H n ( u ( x, t ) − s ).3.1. Uniqueness for first-order flows.
Here we consider the case of curvatures,which generate a first order flow. Namely, we denote by C , the class of sets of R N ,which are the closure of an open set of class C , with compact boundary, and weassume that the following property holds:(FO): Let Σ ∈ C , , let x ∈ ∂ Σ, and let ( p, X ) and ( p, Y ) be elements of J , +Σ ( x )and J , − Σ ( x ), respectively. Then,(3.1) κ ∗ ( x, p, X, Σ) = κ ∗ ( x, p, Y, Σ) . Note that the above assumption postulates that the semicontinuous extensions κ ∗ and κ ∗ are independent of the second derivative variables X and Y , at least on C , -sets. Under these circumstances, we may regard the common value of the quantities in (3.1) as an extension of the definition of curvature to sets of class C , ; i.e., forall Σ ∈ C , we may set κ ( x, Σ) := κ ∗ ( x, p, X, Σ) = κ ∗ ( x, p, Y, Σ)for any ( p, X ) ∈ J , +Σ ( x ) and ( p, Y ) ∈ J , − Σ ( x ).In this situation, the problem becomes similar to the methodology developedin [7] by P. Cardaliaguet. In particular, as show the following Lemma 3.3, ourextensions κ ∗ , κ ∗ correspond here precisely to the extensions h ♯ , h ♭ found in eqn (6),(7) of [7] (with h = − κ ). For completeness, and also because of slight differences (inparticular, we have no sign restriction on our curvatures), we present here completeproofs of uniqueness, which rely as in [7] on Ilmanen’s interposition lemma. Lemma 3.3.
Assume that (FO) holds and let F ∈ M . Then, for any ( p, X ) ∈J , + F ( x ) we have (3.2) κ ∗ ( x, p, X, F ) = sup (cid:8) κ ( x, Σ) : Σ ∈ C , ,F ⊂ Σ , x ∈ ∂ Σ , and p ⊥ ∂ Σ at x (cid:9) . Analogously, for any ( p, Y ) ∈ J , − F ( x ) we have (3.3) κ ∗ ( x, p, Y, F ) = inf (cid:8) κ ( x, Σ) : Σ ∈ C , , ˚Σ ⊂ F, x ∈ ∂ Σ , and p ⊥ ∂ Σ at x (cid:9) . Proof.
We only prove (3.2), the proof of (3.3) being analogous. Denote by κ ( x, p, F )the quantity defined by the right-hand side of (3.2). Clearly, by definition of κ ∗ weimmediately have that κ ∗ ( x, p, X, F ) ≤ κ ( x, p, F ).To prove the opposite inequality, fix ε > C , -set, admissiblefor the definition of κ ( x, p, F ), such that(3.4) κ ( x, Σ) ≥ κ ( x, p, F ) − ε . Moreover, let A ∈ C , admissible for the definition of κ ∗ ( x, p, X, Σ) = κ ( x, Σ), suchthat(3.5) κ ( x, A ) ≥ κ ( x, Σ) − ε . Since A is also admissible for F , we have(3.6) κ ∗ ( x, p, X, F ) ≥ κ ( x, A ) ≥ κ ( x, Σ) − ε ≥ κ ( x, p, F ) − ε, and the conclusion follows from the arbitrariness of ε . (cid:3) We continue with the following lemma, which provides a crucial comparisonproperty between κ ∗ and κ ∗ . Lemma 3.4.
Assume that condition (FO) above holds and let F , G be a closedand an open set, respectively, with compact boundaries and such that F ⊂ G . Let x ∈ ∂F , y ∈ ∂G satisfy (3.7) | x − y | = dist( ∂F, ∂G ) . Then, for all ( p, X ) ∈ J , + F ( x ) and ( p, Y ) ∈ J , − G ( y ) , with p := x − y , we have κ ∗ ( x, p, X, F ) ≥ κ ∗ ( y, p, Y, G ) . ONLOCAL CURVATURE FLOWS 21
Proof.
Exploiting (3.7), we may apply the Ilmanen Interposition Lemma ([21]), tofind a set Σ ∈ C , such that F ⊂ Σ, Σ ⊂ G , and [ x, y ] ∩ ∂ Σ = { ˆ z } , with ˆ z satisfying | x − ˆ z | = dist( ∂F, ∂ Σ) = | y − ˆ z | = dist( ∂G, ∂ Σ) . Here [ x, y ] stands for the segment with endpoints x and y . In particular, F ⊂ Σ+ x − ˆ z with x ∈ ∂ (Σ+ x − ˆ z ) and p ⊥ ∂ (Σ+ x − ˆ z ) at x . Analogously ˚Σ+ y − ˆ z ⊂ G ,with y ∈ ∂ (Σ + y − ˆ z ) and p ⊥ ∂ (Σ + y − ˆ z ) at y . Recalling (3.2) and (3.3), wemay conclude κ ∗ ( x, p, X, F ) ≥ κ ( x, Σ + x − ˆ z ) = κ ( x, Σ + y − ˆ z ) ≥ κ ∗ ( y, p, Y, G ) . (cid:3) Uniqueness of viscosity solutions is a straightforward consequence of the followingComparison Principle, which is the main result of this subsection.
Theorem 3.5 (First Order Comparison Principle) . Assume that condition (FO) holds. Let u ∈ U SC ( R N × [0 , T ]) and v ∈ LSC ( R N × [0 , T ]) , both constant (spatially)out of a compact set, be a subsolution and a supersolution of (2.4) , respectively. If u ( · , ≤ v ( · , , then u ≤ v in R N × [0 , T ] .Proof. Assume by contradiction that there exists z := (¯ x, ¯ t ) ∈ R N × (0 , T ] such that u (¯ z ) − v (¯ z ) >
0. Let f ∈ F , α, ε > x, t, y, s ) := u ( x, t ) − v ( y, s ) − αf ( | x − y | ) − α ( t − s ) − εt − εs. Notice that Φ is u.s.c. Let (ˆ x, ˆ t, ˆ y, ˆ s ) ∈ R N × [0 , T ] × R N × [0 , T ] be a maximumpoint of Φ. We may choose ε so small that the maximum is strictly positive.Moreover, as α → ∞ we clearly have | ˆ x − ˆ y | , | ˆ s − ˆ t | →
0. Thus, since Φ( x, , x, ≤ α so large that ˆ s and ˆ t are strictly positive. We consider now twocases. First case: ˆ x = ˆ y . Let(3.8) ϕ ( x, t ) := v (ˆ y, ˆ s ) + αf ( | x − ˆ y | ) + α ( t − ˆ s ) + εt + ε ˆ s. (3.9) ψ ( y, s ) := u (ˆ x, ˆ t ) − αf ( | ˆ x − y | ) − α (ˆ t − s ) − ε ˆ t − εs. Since u is a subsolution and v is a supersolution we have0 ≥ ϕ t (ˆ x, ˆ t ) = 2 α (ˆ t − ˆ s ) + ε, ≤ ψ t (ˆ y, ˆ s ) = 2 α (ˆ t − ˆ s ) − ε, which yields a contradiction. Second case: ˆ x = ˆ y . Note that (cid:16) α (ˆ t − ˆ s ) + ε, αf ′ ( | ˆ p | ) ˆ p | ˆ p | , X (cid:17) ∈ P , + u (ˆ x, ˆ t ) , (cid:16) α (ˆ t − ˆ s ) − ε, αf ′ ( | ˆ p | ) ˆ p | ˆ p | , − X (cid:17) ∈ P , − v (ˆ y, ˆ s ) , where ˆ p := ˆ x − ˆ y and X := D ϕ (ˆ x, ˆ t ), with ϕ defined in (3.8). Thus, by Lemma 2.15and Remark 2.17, we have(3.10) 2 α (ˆ t − ˆ s ) + ε + k ∗ (cid:16) ˆ x, αf ′ ( | ˆ p | ) ˆ p | ˆ p | , X, { u ( · , ˆ t ) ≥ u (ˆ x, ˆ t ) } (cid:17) ≤ , α (ˆ t − ˆ s ) − ε + k ∗ (cid:16) ˆ y, αf ′ ( | ˆ p | ) ˆ p | ˆ p | , − X, { v ( · , ˆ s ) > v (ˆ y, ˆ s ) } (cid:17) ≥ . Observe now that if x ∈ { u ( · , ˆ t ) ≥ u (ˆ x, ˆ t ) } and | y − x | < | ˆ y − ˆ x | , then v (ˆ y, ˆ s ) − v ( y, ˆ s ) ≤ u (ˆ x, ˆ t ) − u ( x, t ) + αf ( | x − y | ) − αf ( | ˆ x − ˆ y | ) < y ∈ { v ( · , ˆ s ) > v (ˆ y, ˆ s ) } . In other words, { u ( · , ˆ t ) ≥ u (ˆ x, ˆ t ) } + B (0 , | ˆ y − ˆ x | ) ⊂ { v ( · , ˆ s ) > v (ˆ y, ˆ s ) } , which by Lemma 3.4 implies k ∗ (cid:16) ˆ x, αf ′ ( | ˆ p | ) ˆ p | ˆ p | , X, { u ( · , ˆ t ) ≥ u (ˆ x, ˆ t ) } (cid:17) ≥ k ∗ (cid:16) ˆ y, αf ′ ( | ˆ p | ) ˆ p | ˆ p | , − X, { v ( · , ˆ s ) > v (ˆ y, ˆ s ) } (cid:17) . This inequality, combined with (3.10), easily leads to the contradiction 2 ε ≤
0. Thisconcludes the proof of the theorem. (cid:3)
Uniqueness for second-order flows.
Here we consider general second-orderflows. In order to establish uniqueness we will need to assume the following rein-forced continuity property, which replaces C) of Subsection 2.1:C’) Uniform continuity: Given
R >
0, there exists a modulus of continuity ω R such that the following holds. For all E ∈ C , x ∈ ∂E , such that E hasboth an internal and external ball condition of radius R at x , and for allΦ : R N → R N diffeomorphism of class C ℓ,β , with Φ( y ) = y for | y − x | ≥ | κ ( x, E ) − κ (Φ( x ) , Φ( E )) | ≤ ω R ( k Φ − Id k C ℓ,β ) . (We can observe that if Φ is a translation out of B ( x, κ ∗ and κ ∗ . Lemma 3.6.
Assume that C’) holds. Then, given
R > , there exists a modulus ofcontinuity ω R such that the following holds. For all F ∈ M , x ∈ ∂F , with internaland external ball condition at x of radius R , any ( p, X ) ∈ J , + F ( x ) with p = 0 , | X | / | p | ≤ /R , then for any Φ : R N → R N diffeomorphism of class C ℓ,β , we have | κ ∗ ( x, p, X, F ) − κ ∗ (Φ( x ) , D ( f ◦ Φ − )(Φ( x )) , D ( f ◦ Φ − )(Φ( x )) , Φ( F )) |≤ ω R ( k Φ − Id k C ℓ,β ) where f ( y ) = ( y − x ) · p + X ( y − x ) · ( y − x ) . The same holds true for κ ∗ . ONLOCAL CURVATURE FLOWS 23
Proof.
Let E ∈ C with E ⊇ F and ( p, X ) ∈ J , − E ( x ). A first remark is that E hasan inner ball condition at x of radius R , since it contains F . Given R ′ < R , andletting B R ′ = B ( x − R ′ p/ | p | , R ′ ) be the external ball of radius R ′ which touches ∂F at x (only), we observe that we may find a set E ′ ∈ C such that E ′ ⊆ E \ B R ′ and E ′ ⊇ F , and still (since the ball B R ′ is also exterior and tangent to the set { f ≥ } ,thanks to the condition | X | / | p | ≤ /R ), ( p, X ) ∈ J , − E ′ ( x ). By assumption A) onehas κ ( x, E ′ ) ≥ κ ( x, E ). By assumption C’), κ ( x, E ′ ) − κ (Φ( x ) , Φ( E ′ )) ≤ ω R ′ ( k Φ − Id k C ℓ,β ) . Since Φ( E ′ ) ∈ C , Φ( E ′ ) ⊇ Φ( F ) and ( D ( f ◦ Φ − )(Φ( x )) , D ( f ◦ Φ − )(Φ( x ))) ∈J , − Φ( E ′ ) (Φ( x )), then κ ∗ (Φ( x ) , D ( f ◦ Φ − )(Φ( x )) , D ( f ◦ Φ − )(Φ( x )) , Φ( F )) ≥ κ (Φ( x ) , Φ( E ′ )) ≥ κ ( x, E ′ ) − ω R ′ ( k Φ − Id k C ℓ,β ) ≥ κ ( x, E ) − ω R ′ ( k Φ − Id k C ℓ,β ) . Taking the the supremum over all suitable sets E , we deduce κ ∗ (Φ( x ) , D ( f ◦ Φ − )(Φ( x )) , D ( f ◦ Φ − )(Φ( x )) , Φ( F )) ≥ κ ∗ ( x, p, X, F ) − ω R ′ ( k Φ − Id k C ℓ,β ) . The other inequality in order to conclude the proof (possibly redefining slightly ω R )follows analogously, swiching the role of E with Φ( E ). (cid:3) Now, if C’) holds, we also deduce a natural comparison property for the curva-tures κ ∗ , κ ∗ of sets included in one another with a unique contact point. Lemma 3.7.
Assume C’) holds. Let x ∈ R N , F, G ∈ M with F ⊂ G ∪ { x } and ∂F ∩ ∂G = { x } . Let ( p, X ) ∈ J , + F ( x ) , ( p, Y ) ∈ J , − G ( x ) , with X ≤ Y . Then, κ ∗ ( x, p, X, F ) ≥ κ ∗ ( x, p, Y, G ) . Proof.
First we observe that if
X < Y then one may find X ′ , Y ′ with X < X ′
Theorem 3.8 (Second Order Comparison Principle) . Assume C’) holds. Let u ∈ U SC ( R N × [0 , T ]) and v ∈ LSC ( R N × [0 , T ]) , both constant (spatially) out of acompact set, be a subsolution and a supersolution of (2.4) , respectively. If u ( · , ≤ v ( · , , then u ≤ v in R N × [0 , T ] .Proof. Without loss of generality we can assume u ( · , < v ( · , a ∈ R and t ∈ (0 , T ] such that F ( t ) := { u ( · , t ) ≥ a } isnot contained in G ( t ) := { v ( · , t ) > a } .Notice that, since u ( · , < v ( · , F (0) ⊂ G (0). A first remark is thatwe can assume, without loss of generality, that the sets F ( t ) satisfy an internal ballcondition with some fixed radius r >
0, at all time, while G ( t ) satisfy an externalball condition with same radius. Indeed, we can always replace F and G with,respectively, the sets F ( t ) + B (0 , r ) = [ | z |≤ r ( z + F ( t )) and { x : B ( x, r ) ⊂ G ( t ) } for some r >
0, which are still, respectively, a sub and a super-solution of (1.1);moreover if r is small enough, the inclusion F (0) ⊂ G (0) is preserved.Let f ∈ F ∩ C ℓ,β , and let u λF ( x, t ) := max ξ ∈ R N ,τ ∈ [ t − T,t ] χ F ( t − τ ) ( x − ξ ) − λ ( f ( | ξ | ) + τ ) , (3.11) v λG ( x, t ) := min ξ ∈ R N ,τ ∈ [ t − T,t ] χ G ( t − τ ) ( x − ξ ) + λ ( f ( | ξ | ) + τ )(3.12)Since F (0) ⊂ G (0), for λ large enough u λF ( · , ≤ v λG ( · , x, t, y, s ) := u λF ( x, t ) − v λG ( y, s ) − ε ( t + s ) − λ ( f ( | x − y | ) + | t − s | )is semiconvex, and for ε > λ large enough admits a positive maximumat some ( x , t , y , s ) ∈ R N × [0 , T ] × R N × [0 , T ] with t , s > First case: x = y . Let ϕ ( x, t ) := v λG ( y , s ) + λf ( | x − y | ) + λ ( t − s ) + εt + εs .ψ ( y, s ) := u λF ( x , t ) − λf ( | x − y | ) − λ ( t − s ) − εt − εs. We observe that by construction, u λF is a subsolution and v λG is a supersolution on R N × [2 / √ λ, T ] (we need t ≥ / √ λ to ensure that the max in (3.11) is not reachedat τ = t , observe though that if λ is large enough one will have t , s > / √ λ ).Hence, we have0 ≥ ϕ t ( x , t ) = 2 λ ( t − s ) + ε, ≤ ψ t ( y , s ) = 2 λ ( t − s ) − ε, which yield a contradiction. ONLOCAL CURVATURE FLOWS 25
Second case: x = y . We can always assume (choosing λ large enough) that f ( | x − y | ) <
1. Moreover, we have(3.13) u λF ( x , t ) < . Indeed, observe that u λF ( x, t ) = 1 if and only if x ∈ F ( t ), that Du λF ( x, t ) = 0 on F ( t ), and Du λF ( x , t ) = D x f ( | x − y | ) = 0. Thus, x can not belong to F ( t ).Let q : [0 , + ∞ ] → [0 ,
1] be a smooth, nondecreasing, function with q ( r ) = r for r < / q ( r ) = 1 for r > /
2. For ρ >
0, let thenΦ ρ ( x, t, y, s ) := Φ( x, t, y, s ) − ρ [ q ( | x − x | ) + q ( | y − y | ) + q ( | t − t | ) + q ( | s − s | )] , so that ( x , t , y , s ) is a strict maximum of Φ ρ . Let η : R N → R be a smoothcut-off function, with compact support and equal to one in a neighborhood U ofthe origin. For every ∆ := ( ζ u , h u , ζ v , h v ) ∈ R N × R × R N × R , the functionΦ ρ ( x, t, y, s ) − (cid:16) η ( x − x )( ξ u , h u ) · ( x, t ) + η ( y − y )( ξ v , h v ) · ( y, s ) (cid:17) is maximized at some ( x ∆ , t ∆ , y ∆ , s ∆ ) such that( x ∆ , t ∆ , y ∆ , s ∆ ) → ( x , t , y , s ) as | ∆ | → . Thus, by Jensen’s Lemma [16, Lemma A.3], we may assume that for every δ > ρ,δ := ( ζ ρ,δu , h ρ,δu , ζ ρ,δv , h ρ,δv ), with | ∆ ρ,δ | ≤ δ , suchthat the functionΦ ρ,δ ( x, t, y, s ) := Φ ρ ( x, t, y, s ) − (cid:16) η ( x − x )( ξ ρ,δu , h ρ,δu ) · ( x, t ) + η ( y − y )( ξ ρ,δv , h ρ,δv ) · ( y, s ) (cid:17) attains a maximum at some z ρ,δ := ( x ρ,δ , t ρ,δ , y ρ,δ , s ρ,δ ) where Φ δ,ρ is twice differ-entiable and such that x ρ,δ − x , y ρ,δ − y ∈ U and t ρ,δ , s ρ,δ >
0. Moreover,(3.14) z ρ,δ → ( x , t , y , s ) as δ → . Notice that since Φ ρ is twice differentiable at z ρ,δ it follows that also u λF and v λG aretwice differentiable at ( x ρ,δ , t ρ,δ ) and ( y ρ,δ , s ρ,δ ), respectively.Let τ ρ,δu , τ ρ,δv ∈ R be the maximizing τ ’s in (3.11), (3.12) corresponding to thepoints ( x ρ,δ , t ρ,δ ), ( y ρ,δ , s ρ,δ ), respectively. Set˜ u F ( x, t ) := max ξ ∈ R N n χ F ( t − τ ρ,δu ) ( x − ξ ) − λf ( | ξ | ) o − λ ( τ ρ,δu ) (3.15) ˜ v G ( y, s ) := min ξ ∈ R N n χ G ( s − τ ρ,δv ) ( y − ξ ) + λf ( | ξ | ) o + λ ( τ ρ,δv ) . (3.16)Observe that by construction, u λF ≥ ˜ u F , v λG ≤ ˜ v G , (3.17) u λF ( x ρ,δ , t ρ,δ ) = ˜ u F ( x ρ,δ , t ρ,δ ) , v λG ( y ρ,δ , s ρ,δ ) = ˜ v G ( y ρ,δ , s ρ,δ ) . Set nowˆ u F ( x, t ) := ˜ u F ( x, t ) − ρ q ( | x − x ρ,δ | ) − ρ [ q ( | x − x | ) + q ( | t − t | )] − η ( x − x )( ξ ρ,δu , h ρ,δu ) · ( x, t ) , ˆ v G ( y, s ) := ˜ v G ( y, s ) + ρ q ( | y − y ρ,δ | )+ ρ [ q ( | y − y | ) + q ( | s − s | )] + η ( y − y )( ξ ρ,δv , h ρ,δv ) · ( y, s ) . Then by construction, the functionˆ u F ( x, t ) − ˆ v G ( y, s ) − ε ( t + s ) − λ ( f ( | x − y | ) + | t − s | ) . has its maximum at z ρ,δ , which is now strict with respect to the spatial variables.Hence if we setˆ F ρ,δ ( t ) := { ˆ u F ( · , t ) ≥ ˆ u F ( x ρ,δ , t ρ,δ ) } , ˆ G ρ,δ ( s ) := { ˆ v G ( · , s ) > ˆ v G ( y ρ,δ , s ρ,δ ) } . we have ˆ F ρ,δ ( t ρ,δ ) ⊆ ˆ G ρ,δ ( s ρ,δ ) and moreover ( x ρ,δ , y ρ,δ ) is the only pair of minimaldistance in ∂ ˆ F ρ,δ ( t ρ,δ ) × ∂ ˆ G ρ,δ ( s ρ,δ ). In addition, we observe that at the maximumpoint, | D ˆ u F ( x ρ,δ , t ρ,δ ) | ≈ | D ˜ u ( x ρ,δ , t ρ,δ ) | = | Du λF ( x ρ,δ , t ρ,δ ) | ≈ λf ′ ( | x ρ,δ − y ρ,δ | ) ≈ λf ′ ( | x − y | ) = 0 up to perturbations which go to zero as ρ, δ →
0, and that thefunction ˆ u F is semiconvex, hence ˆ F ρ,δ ( t ρ,δ ) has an interior ball condition at x ρ,δ with a radius independent on ρ and δ , if small enough. In the same way, ˆ G ρ,δ ( s ρ,δ )has an exterior ball condition at y ρ,δ . In turns, ˆ F ρ,δ ( t ρ,δ ) and ˆ G ρ,δ ( s ρ,δ ) satisfy bothand internal and external ball condition.Set ˘Φ ρ,δ ( x, t, y, s ) := Φ ρ,δ ( x, t, y, s ) + λ ( f ( | x − y | ) + | t − s | )and (˘ a ρ,δ , ˘ p ρ,δ , ˘ X ρ,δ ) := ( ∂ t ˘Φ ρ,δ ( z ρ,δ ) , D x ˘Φ ρ,δ ( z ρ,δ ) , D x ˘Φ ρ,δ ( z ρ,δ )) , (3.18) (˘ b ρ,δ , ˘ q ρ,δ , ˘ Y ρ,δ ) := ( ∂ s ˘Φ ρ,δ ( z ρ,δ ) , D y ˘Φ ρ,δ ( z ρ,δ ) , D y ˘Φ ρ,δ ( z ρ,δ )) . (3.19)Then, recalling (3.17), we observe that the superjet (˘ a ρ,δ , ˘ p ρ,δ , ˘ X ρ,δ ) of u λF ( x, t ) − ρ [ q ( | x − x | ) + q ( | t − t | )] − η ( x − x )( ξ ρ,δu , h ρ,δu ) · ( x, t )at ( x ρ,δ , t ρ,δ ) is also a superjet for ˆ u F ( x, t ) at the same point, hence, also, for thefunction ˆ u F ( x ρ,δ , t ρ,δ ) χ ˆ F ρ,δ (since ˆ u F ( x, t ) ≥ ˆ u F ( x ρ,δ , t ρ,δ ) χ ˆ F ρ,δ ( t ) ( x )). Hence, wehave(3.20) (˘ a ρ,δ , ˘ p ρ,δ , ˘ X ρ,δ ) ∈ P , +ˆ u F ( x ρ,δ ,t ρ,δ ) χ ˆ Fρ,δ ( x ρ,δ , t ρ,δ )and analogously, (˘ b ρ,δ , ˘ q ρ,δ , ˘ Y ρ,δ ) ∈ P , − ˆ v G ( y ρ,δ ,s ρ,δ ) χ ˆ Gρ,δ ( y ρ,δ , s ρ,δ ) . Since D z Φ ρ,δ ( z ρ,δ ) = 0, we deduce(3.21) ˘ a ρ,δ − ˘ b ρ,δ = 2 ε, ˘ p ρ,δ = ˘ q ρ,δ . Moreover, since D x,y Φ ρ,δ ( z ρ,δ ) ≤
0, one can check that(3.22) ˘ X ρ,δ ≤ ˘ Y ρ,δ . By construction, ˘Φ ρ,δ is also semiconvex, so that ˘ X ρ,δ ≥ − cI , ˘ Y ρ,δ ≤ cI for aconstant c depending on λ .Let c ρ,δ ( x, t ) := ˜ u F ( x, t ) + (ˆ u F ( x ρ,δ , t ρ,δ ) − ˆ u F ( x, t )) . ONLOCAL CURVATURE FLOWS 27
Notice that, as δ → ρ →
0, we have c ρ,δ → u λF ( x , t ) uniformly. In view of (3.13)we can thus assume that c ρ,δ <
1. Observe also that c ρ,δ is smooth and constantaway from a neighborhood of ( x , t ), and it converges also in C ℓ,β .We have ˆ F ρ,δ ( t ) = { x : ˜ u F ( x, t ) ≥ c ρ,δ ( x, t ) } . Thus, by the definition of ˜ u F , wehave that x ∈ ˆ F ρ,δ ( t ) if and only if there exists ξ ∈ R N such that x ∈ ξ + F ( t − τ ρ,δu ),with χ ξ + F ( t − τ ρ,δu ) ( x ) − λf ( | ξ | ) = 1 − λf ( | ξ | ) ≥ c ρ,δ ( x, t ) , i.e.,(3.23) ˆ F ρ,δ ( t ) = ( x : x ∈ ξ + F ( t − τ ρ,δu )for some ξ ∈ R N with | ξ | ≤ f − (cid:18) − c ρ,δ ( x, t ) λ (cid:19)) . For ρ, δ small enough, x ρ,δ F ( t ρ,δ − τ ρ,δu ) and we can introduce w ρ,δ = 0 suchthat x ρ,δ + w ρ,δ is a projection of x ρ,δ on F ( t ρ,δ − τ ρ,δu ). Precisely, one has that c ρ,δ ( x ρ,δ , t ρ,δ ) = u λF ( x ρ,δ , t ρ,δ ), ξ = − w ρ,δ reaches the max in (3.11) (for x = x ρ,δ ),and | w ρ,δ | = f − ((1 − c ρ,δ ( x ρ,δ , t ρ,δ )) /λ ). We then setΨ ρ,δ ( x ) := x − f − (cid:18) − c ρ,δ ( x, t ρ,δ ) λ (cid:19) w ρ,δ | w ρ,δ | + w ρ,δ which is a C ℓ,β diffeomorphism (being c ρ,δ bounded away from 1), which is a con-stant (small) translation out of a neighborhood of x , and converges C ℓ,β to theidentity as δ → ρ →
0. Observe that Ψ ρ,δ ( x ρ,δ ) = x ρ,δ . Then, we let(3.24) ˇ F ρ,δ ( t ) := Ψ ρ,δ ( F ( t − τ ρ,δu ) − w ρ,δ ) . By construction, ˇ F ρ,δ ( t ρ,δ ) ⊆ ˆ F ρ,δ ( t ρ,δ ) and x ρ,δ ∈ ∂ ˇ F ρ,δ ( t ρ,δ ) ∩ ∂ ˆ F ρ,δ ( t ρ,δ ). More-over, we recall that ˆ F ρ,δ ( t ρ,δ ) has an internal ball condition at x ρ,δ while ˆ G ρ,δ ( s ρ,δ )satisfies an external ball condition at y ρ,δ , with radius bounded away from 0 uni-formly with respect to ρ and δ sufficiently small. Thus, recalling that ˆ F ρ,δ ( t ρ,δ ) +( y ρ,δ − x ρ,δ ) ⊆ ˆ G ρ,δ ( s ρ,δ ) (being y ρ,δ the only contact point), we have that ˆ F ρ,δ ( t ρ,δ ),and in turn ˇ F ρ,δ ( t ρ,δ ) satisfies a uniform external ball condition in x ρ,δ . In addition,since we have assumed that F ( t ) had a uniform internal ball condition for someradius r >
0, the same holds for ˇ F ρ,δ ( t ρ,δ ) with a smaller radius.Notice that k Ψ ρ,δ − I k C ℓ,β → ρ, δ →
0. Set( p ρ,δ , X ρ,δ ) := (cid:16) D x ( ˘Φ ρ,δ ( · , t ρ,δ , y ρ,δ , s ρ,δ ) ◦ Ψ ρ,δ )( x δ ) ,D x ( ˘Φ ρ,δ ( · , t ρ,δ , y ρ,δ , s ρ,δ ) ◦ Ψ ρ,δ )( x δ ) (cid:17) , By construction (see (3.20)), we have(˘ a ρ,δ , p ρ,δ , X ρ,δ ) ∈ P , +ˆ u F ( x ρ,δ ,t ρ,δ ) χ F ( t − τρ,δu ) ( x ρ,δ + w ρ,δ )Since ˆ u F ( x ρ,δ , t ρ,δ ) χ F ( t − τ ρ,δu ) is a subsolution, we have(3.25) ˘ a ρ,δ + | p ρ,δ | κ ∗ ( x ρ,δ + w ρ,δ , p ρ,δ , X ρ,δ , F ( t ρ,δ − τ ρ,δu )) ≤ . Note that p ρ,δ → Du λF ( x , t ) = 0 , as ρ , δ →
0, and thus | p ρ,δ | is bounded away from zero for ρ and δ sufficiently small.Since also ˘ X ρ,δ and hence X ρ,δ is bounded, we can invoke Lemma 3.6 and deducethat(3.26) ˘ a ρ,δ + | ˘ p ρ,δ | κ ∗ ( x ρ,δ , ˘ p ρ,δ , ˘ X ρ,δ , ˇ F ρ,δ ( t ρ,δ )) ≤ ω ( ρ, δ ) , where ω ( ρ, δ ) → ρ, δ → a ρ,δ − ε + | ˘ p ρ,δ | κ ∗ ( y ρ,δ , ˘ p ρ,δ , ˘ Y ρ,δ , ˇ G ρ,δ ( s ρ,δ )) ≥ ω ( ρ, δ )for a suitable set ˇ G ρ,δ ( s ρ,δ )) such that ˆ F ( t ρ,δ ) + ( y ρ,δ − x ρ,δ ) ⊆ ˇ G ρ,δ ( s ρ,δ )) and ∂ ( ˇ F ρ,δ ( t ρ,δ ) + ( y ρ,δ − x ρ,δ )) ∩ ∂ ˇ G ρ,δ ( s ρ,δ )) = { y ρ,δ } . By (3.22) and Lemma 3.7 weget κ ∗ ( x ρ,δ , ˘ p ρ,δ , ˘ X ρ,δ , ˇ F ρ,δ ( t ρ,δ )) ≥ κ ∗ ( y ρ,δ , ˘ p ρ,δ , ˘ Y ρ,δ , ˇ G ρ,δ ( s ρ,δ )) , and thus, in particular,˘ a ρ,δ − ε + | ˘ p ρ,δ | κ ∗ ( x ρ,δ , ˘ p ρ,δ , ˘ X ρ,δ , ˇ F ρ,δ ( t ρ,δ )) ≥ ω ( ρ, δ ) , which, together (3.26) gives ε ≤ ω ( ρ, δ ). This is a contradiction for ρ, δ sufficientlysmall. (cid:3) Remark 3.9.
By the uniqueness property stated in Theorem 3.5 and Theorem3.8, we deduce that the evolution of open and closed superlevel sets is intrinsicin the following sense. Let u , ˜ u : R N → R be two initial conditions such that { u ( · ) > } = { ˜ u ( · ) > } . Then, denoting by u and ˜ u the corresponding geometricevolutions we have { u ( · , t ) > } = { ˜ u ( · , t ) > } for all t ∈ [0 , t ](and the same identity holds for the closed superlevels). Indeed, for any λ ≥ A λ ( t ) := { u ( · , t ) > λ } , ˜ A λ ( t ) := { ˜ u ( · , t ) > λ } ,C λ ( t ) := { u ( · , t ) ≥ λ } , ˜ C λ ( t ) := { ˜ u ( · , t ) ≥ λ } , In view of Lemma 3.2 and of Theorems 3.5 and 3.8, we have that for every λ > C λ ( t ) ⊆ A ( t ) , C λ ( t ) ⊆ ˜ A ( t ) . Thus, we conclude˜ A ( t ) = [ λ> ˜ C λ ( t ) ⊆ A ( t ) , A ( t ) = [ λ> C λ ( t ) ⊆ ˜ A ( t ) . ONLOCAL CURVATURE FLOWS 29
Part Variational Nonlocal Curvature Flows
In this part we further assume the nonlocal curvature to be variational ; thatis, we assume that κ is the first variation of a suitable generalized perimeter . Thesecond part of the paper is organized as follows. In Section 4 we introduce a suitableclass of translation invariant generalized perimeters J and we give a rather weaknotion of curvature as a first variation of the perimeter functional with respect tomeasurable perturbations of the set shrinking to a point x of the boundary. Wealso show how some of the structural assumptions of J translate into properties ofthe curvature κ .In Section 5 we study how the weak notion of curvature compares to more stan-dard ones. Section 6 is devoted to showing some relevant examples of variationalnonlocal curvatures that fit in our abstract framework. Finally, Section 7 containsthe main result of this part, namely the fact that the minimizing movement schemeapplied to J converges to the associated nonlocal curvature flow.4. Generalized perimeters and curvatures
Generalized perimeters.
We now will consider a generalized notion of (pos-sibly nonlocal) perimeter. We will say that a functional J : M → [0 , + ∞ ] is ageneralized perimeter if it satisfies the following properties:i) J ( E ) < + ∞ for every E ∈ C ;ii) J ( ∅ ) = J ( R N ) = 0;iii) J ( E ) = J ( E ′ ) if | E △ E ′ | = 0;iv) J is L loc -l.s.c.: if | ( E n △ E ) ∩ B R | → R >
0, then J ( E ) ≤ lim inf n J ( E n );v) J is submodular: For any E, F ∈ M ,(4.1) J ( E ∪ F ) + J ( E ∩ F ) ≤ J ( E ) + J ( F ) ;vi) J is translational invariant:(4.2) J ( x + E ) = J ( E ) for all E ∈ M , x ∈ R N . We can extend the functional J to L loc ( R N ) enforcing the following generalizedco-area formula :(4.3) J ( u ) := Z + ∞−∞ J ( { u > s } ) ds for every u ∈ L loc ( R N ) . It can be shown that, under the assumptions above, J is a convex l.s.c. functionalin L loc ( R N ) (see [13]).Observe that the following holds true: Lemma 4.1.
Let u ∈ L loc ( R N ) and ρ be a nonnegative and compactly supportedmollifier, and let ρ ε ( x ) := ρ ( x/ε ) /ε N for ε > small. Then for all ε , (4.4) J ( ρ ε ∗ u ) ≤ J ( u ) . Moreover, lim ε → J ( ρ ε ∗ u ) = J ( u ) . Proof.
The first statement follows from the convexity of J , by approximating ρ ε ∗ u by appropriate finite convex combinations and then passing to the limit thanks tothe lower semicontinuity assumption iv). The last statement is then an immediateconsequence of (4.4) and once again of assumption iv). (cid:3) A weak notion of curvature.
We introduce here a definition of the curvatureof sets relative to the generalized perimeter J which will be useful for studying thegeometric “gradient flow” of J . It is based on a sort of local subdifferentiabilityproperty. We will show in Section 5 that it is implied by more standard definitionsbased on global variations of the boundaries of smooth sets. Definition 4.2.
Let E ∈ C and x ∈ ∂E . We set(4.5) κ + ( x, E ) := inf (cid:26) lim inf n J ( E ∪ W n ) − J ( E ) | W n \ E | : W n H → { x } , | W n \ E | > (cid:27) and(4.6) κ − ( x, E ) := sup (cid:26) lim sup n J ( E ) − J ( E \ W n ) | W n ∩ E | : W n H → { x } , | W n ∩ E | > (cid:27) . We say that κ ( x, E ) is the curvature of E at x (associated with the perimeter J ) if κ + ( x, E ) = κ − ( x, E ) =: κ ( x, E ) ∈ R .Notice that if J ( E ) = J ( R N \ E ) it follows that κ + ( x, E ) = − κ − ( x, R N \ E ), andtherefore κ ( x, E ) = − κ ( x, R N \ E ) (whenever it exists). Standing Assumptions of Part 2 . Throughout Part 2 we assume that thecurvature exists for all sets in C , i.e., κ ( x, E ) := κ + ( x, E ) = κ − ( x, E ) ∈ R for all E ∈ C and all x ∈ ∂E ,and that κ satisfies axiom C) of Subsection 2.1. The translational invariance B) follows naturally from the translational invari-ance of the perimeter J . The monotonicity property A) stated in Subsection 2.1 isa consequence of the submodularity assumption (4.1), as shown in the next lemma. Lemma 4.3.
Let
E, F ∈ C with E ⊆ F , and assume that x ∈ ∂F ∩ ∂E : then κ ( x, F ) ≤ κ ( x, E ) .Proof. First, we observe that we can find sets F n which converge to F in C , with F n ⊇ F and { x } = ∂E ∩ ∂F n . In particular, κ ( x, F n ) → κ ( x, F ). Then, let ν bethe (outer) normal vector to E and (all of the) F n at x , and for ε > E ε = E + εν . Let W ε = E ε \ ˚ F n , and observe that W ε H → { x } as ε →
0, and | W ε | > ε > E ε and F n ) and (4.2), wehave J ( F n ∪ W ε ) − J ( F n ) ≤ J ( E ε ) − J (( E ε ) \ W ε ) = J ( E ) − J ( E \ ( W ε − εν )) . Then, by the very definition of κ we deduce κ ( x, F n ) ≤ κ ( x, E ). The conclusionfollows noticing that, by the continuity property C), κ ( x, F n ) → κ ( x, F ). (cid:3) ONLOCAL CURVATURE FLOWS 31 First variation of the perimeter
Let J be a generalized perimeter. In this section we compare the weak notion ofcurvature given in Definition 4.2 with the more standard one based on the first vari-ation of the perimeter functional. The latter is in turn related to shape derivatives ,a notion which dates back to Hadamard, extensively studied in particular in [26]. Definition 5.1.
We say that κ ( x, E ), defined for E ∈ C and x ∈ ∂E , is the firstvariation of the perimeter J if for every E ∈ C , and any one-parameter family ofdiffeomorphisms (Φ ε ) ε of class C ℓ,β both in x and in ε with Φ ( x ) = x , we have(5.1) ddε J (cid:0) Φ ε ( E ) (cid:1) | ε =0 = Z ∂E κ ( x, E ) ψ ( x ) · ν E ( x ) d H N − ( x ) , where ψ ( x ) := ∂ Φ ε ∂ε ( x ) | ε =0 and ν E ( x ) is the C ℓ − ,β outer normal to the set E at x .We will show that if such a κ ( x, E ) is continuous with respect to C ℓ,β perturba-tions of the sets E , then it is also a curvature in the sense of Definition 4.2. Westart with the following intermediate result. Proposition 5.2.
Let κ ( x, E ) be a function defined for all E ∈ C and x ∈ ∂E , andassume that it satisfies the continuity property C). Then κ ( x, E ) is a first variationof J in the sense of Definition 5.1 if and only if for any ϕ ∈ C ℓ,βc ( R N ) , and any t < t such that Dϕ = 0 in the set { t ≤ ϕ ≤ t } , one has (5.2) J ( { ϕ ≥ t } ) = J ( { ϕ ≥ t } ) + Z { t <ϕ Consider the function f ( s ) := J ( { ϕ ≥ s } ), for t < s < t . We claim that if κ is a first variation of J , then(5.4) f ′ ( s ) = − Z ∂ { f ≥ s } κ ( x, { ϕ ≥ s } ) | Dϕ ( x ) | d H N − ( x )for all s ∈ ( t , t ). To this aim, given s ∈ ( t , t ) with t < s < t , we need tofind a family of C ℓ,β diffeomorphisms which transport { ϕ ≥ s } on { ϕ ≥ s + ε } andcompute its derivative at ε = 0.If ϕ were smooth (at least C ℓ +1 ,β ), a simple way would be to consider a smoothvector field V ( x ) which is zero in { ϕ ≥ t } ∪ { ϕ ≤ t } and equal to Dϕ/ | Dϕ | in aneighborhood of { ϕ = s } . We would then let Φ ε ( x ) defined for all x by d Φ ε ( x ) dε = V (Φ ε ( x )) ε > , Φ ( x ) = x. In this case for ε small, we would have that ϕ ( x ) = s implies ϕ (Φ ε ( x )) = s + ε ,since clearly d ( ϕ (Φ ε ( x ))) /dε = Dϕ (Φ ε ( x )) · V (Φ ε ( x )) = 1 for such x and ε . Then, (5.4) would follow from (5.1). However, if ϕ is merely C ℓ,β , this construction buildsonly a C ℓ − ,β diffeomorphism.In general the situation is a bit more complex, however it is clear that such adiffeomorphism exists. A relatively simple construction consists in smoothing ϕ with a smooth mollifier in order to find a C ∞ set ˜ E such that for all ε small enough(here both positive and nonpositive), the surfaces ∂ { ϕ ≥ s + ε } are represented as C ℓ,β graphs over ∂ ˜ E : ∂ { ϕ ≥ s + ε } = n x + h ε ( x ) ν ˜ E ( x ) : x ∈ ∂ ˜ E o . By the implicit function theorem, h ε exists and is C ℓ,β (in both ε and x ) for ε near0. Moreover, since ϕ ( x + h ε ( x ) ν ˜ E ( x )) = s + ε , one checks that for any ε small and x ∈ ∂ ˜ E , ∂h ε ( x ) ∂ε = 1 Dϕ ( x + h ε ( x ) ν ˜ E ( x )) · ν ˜ E ( x ) . The diffeomorphism Φ ε ( x ) is then simply defined, in a neighborhood of the surface ∂ ˜ E , by(5.5) Φ ε ( x ) = x + ( h ε ( π ∂ ˜ E ( x )) − h ( π ∂ ˜ E ( x ))) ν ˜ E ( π ∂ ˜ E ( x )) , where π ∂ ˜ E denotes the orthogonal projection onto ∂ ˜ E , which is well-defined andsmooth in a sufficiently small neighborhood of the surface.Then, one has that for x ∈ ∂ { ϕ ≥ s } (which is the graph of h ) ψ ( x ) := lim ε → Φ ε ( x ) − xε = ∂h ε ∂ε | ε =0 ( π ∂ ˜ E ( x )) ν ˜ E ( π ∂ ˜ E ( x )) = ν ˜ E ( π ∂ ˜ E ( x )) Dϕ ( x ) · ν ˜ E ( π ∂ ˜ E ( x ))and, in turn, ψ ( x ) · ν ∂ { ϕ ≥ s } ( x ) = − ψ ( x ) · Dϕ | Dϕ | ( x ) = − | Dϕ ( x ) | . Hence (5.1) yields(5.6) lim ε → J ( { ϕ ≥ s + ε } ) − J ( { ϕ ≥ s } ) ε = lim ε → J (Φ ε ( { ϕ ≥ s } )) − J ( { ϕ ≥ s } ) ε = − Z ∂ { ϕ ≥ s } κ ( x, { ϕ ≥ s } ) | Dϕ ( x ) | d H N − ( x ) , which shows (5.4). Equation (5.2) follows from (5.4) and the co-area formula for BV functions.Conversely, assume now that (5.2) holds for all ϕ ∈ C ℓ,βc ( R N ) and t < t suchthat Dϕ = 0 in the set { t ≤ ϕ ≤ t } . We consider a family of diffeomorphism Φ ε as in Definition 5.1. We start by showing that (5.1) holds. Write E as E = { ϕ ≥ } for a suitable ϕ ∈ C ℓ,β ( R N ), constant out of a compact set and with Dϕ = 0 in { ≤ ϕ ≤ } . Since Φ ε ( E )∆ E is contained in the M ε -neighborhood ( ∂E ) Mε of ∂E for some M > 0, if ε > e Φ ε such that e Φ ε = Id outside ( ∂E ) Mε (and in particular out of {| ϕ − / | ≤ / } ), e Φ ε ( E ) = Φ ε ( E ), and k e Φ ε − Id k C ℓ,β → ε → 0. In particular, by construction J (Φ ε ( E )) = J ( e Φ ε ( E )). Since Φ ε ( E ) = e Φ ε ( E ) = { ϕ ◦ e Φ − ε ≥ } and { ϕ ◦ e Φ − ε ≥ ONLOCAL CURVATURE FLOWS 33 } = { ϕ ≥ } , by (5.2) we have J (Φ ε ( E )) − J ( { ϕ ≥ } )= Z e Φ ε ( E ) \{ ϕ ≥ } k ( x, E ϕ ◦ e Φ − ε ( x ) ) dx = Z ( ∂E ) Mε ∩ e Φ ε ( E ) (cid:16) k ( x, E ϕ ◦ e Φ − ε ( x ) ) − k ( x, E ϕ ( x ) ) (cid:17) dx + Z e Φ ε ( E ) \{ ϕ ≥ } k ( x, E ϕ ( x ) ) dx ≤ Z ( ∂E ) Mε (cid:12)(cid:12)(cid:12) k ( x, E ϕ ◦ e Φ − ε ( x ) ) − k ( x, E ϕ ( x ) ) (cid:12)(cid:12)(cid:12) dx + Z Φ ε ( E ) \{ ϕ ≥ } k ( x, E ϕ ( x ) ) dx . Since k k ( x, E ϕ ◦ e Φ − ε ( x ) ) − k ( x, E ϕ ( x ) ) k L ∞ (( ∂E ) Mε ) → ε → 0, thanks to Assump-tion C), we have that Z ( ∂E ) Mε (cid:12)(cid:12) k ( x, E ϕ ◦ e Φ − ε ( x ) ) − k ( x, E ϕ ( x ) ) (cid:12)(cid:12) dx = o ( ε ) . Therefore ddε J (cid:0) Φ ε ( E ) (cid:1) | ε =0 = ddε (cid:18)Z Φ ε ( E ) \{ ϕ ≥ } k ( x, E ϕ ( x ) ) dx (cid:19) | ε =0 = Z ∂E k ( x, E ϕ ( x ) ) ψ ( x ) · ν E ( x ) d H N − . It now remains to prove (5.3). We first consider a C ℓ,β function ψ such that ψ − ϕ is compactly supported in { t < ϕ < t } . In particular if ε > { t < ϕ + ε ( ψ − ϕ ) < t } = { t < ϕ < t } . We also introduce ˜ ϕ := t ∨ ( ϕ ∧ t )and ˜ ψ ( x ) := ψ ( x ) if t < ϕ < t , ˜ ψ ( x ) = ˜ ϕ ( x ) ∈ { t , t } else. Observe that thanksto (4.3),(5.7) J ( ˜ ϕ ) = Z t t J ( { ϕ ≥ s } ) ds (which is finite thanks to (5.2)), so that (for ε small) J ( ˜ ϕ + ε ( ˜ ψ − ˜ ϕ )) − J ( ˜ ϕ ) ε = Z t t J ( { ϕ + ε ( ψ − ϕ ) ≥ s } ) − J ( { ϕ ≥ s } )) ε ds. Again, for a fixed s ∈ ( t , t ), one can find a family of C ℓ,β -diffeomorphisms (Φ ε ) ε> which transform { ϕ > s } into { ϕ + ε ( ψ − ϕ ) > s } . One proceeds as before, but nowthe implicit function theorem is applied to the function( x, ε, h ) (1 − ε ) ϕ ( x + hν ˜ E ( x )) + εψ ( x + hν ˜ E ( x )) − s. for x ∈ ∂ ˜ E , ε small enough and h in a suitable neighborhood of 0. The diffeomor-phisms are defined as in (5.5), and we can compute again the derivative of h ε withrespect to ε , for x ∈ ∂ ˜ E : ∂h ε ( x ) ∂ε = ϕ ( x + h ε ( x ) ν ˜ E ( x )) − ψ ( x + h ε ( x ) ν ˜ E ( x ))((1 − ε ) Dϕ ( x + h ε ( x ) ν ˜ E ( x )) + ε ( Dψ ( x + h ε ( x ) ν ˜ E ( x )))) · ν ˜ E ( x ) . Hence, at ε = 0, for x ∈ ∂ { ϕ ≥ s } we have ∂h ε ∂ε | ε =0 ( π ˜ E ( x )) = ϕ ( x ) − ψ ( x ) Dϕ ( x ) · ν ˜ E ( π ˜ E ( x )) . In turn, ψ ( x ) = lim ε → Φ ε ( x ) − xε = ∂h ε ∂ε | ε =0 ( π ∂ ˜ E ( x )) ν ˜ E ( π ∂ ˜ E ( x )) = ( ϕ ( x ) − ψ ( x )) ν ˜ E ( π ∂ ˜ E ( x )) Dϕ ( x ) · ν ˜ E ( π ˜ E ( x )) and ψ ( x ) · ν ∂ { ϕ ≥ s } ( x ) = ψ ( x ) − ϕ ( x ) | Dϕ ( x ) | . Using (5.1), we inferlim ε → J ( { ϕ + ε ( ψ − ϕ ) ≥ s } ) − J ( { ϕ ≥ s } )) ε = Z ∂ { ϕ ≥ s } ψ ( x ) − ϕ ( x ) | Dϕ ( x ) | κ ( x, E ϕ ( x ) ) d H N − ( x ) . Notice that more generally, if ε is small enough, one obtains thatlim ε ′ → ε J ( { ϕ + ε ′ ( ψ − ϕ ) ≥ s } ) − J ( { ϕ + ε ( ψ − ϕ ) ≥ s } )) ε ′ − ε = Z ∂ { ϕ + ε ( ψ − ϕ ) ≥ s } ψ ( x ) − ϕ ( x ) | (1 − ε ) Dϕ ( x ) + εDψ ( x ) | κ ( x, { ϕ + ε ( ψ − ϕ ) ≥ s } ) d H N − ( x ) . Denoting E s,ε the sets { ϕ + ε ( ψ − ϕ ) ≥ s } and observing that they are continuousin C as ε varies, we see that, thanks to assumption C), this derivative is continuouswith respect to ε (small) and in particular,(5.8) J ( { ϕ + ε ( ψ − ϕ ) ≥ s } ) − J ( { ϕ ≥ s } )) ε = 1 ε Z ε Z ∂E s,t ψ ( x ) − ϕ ( x ) | (1 − t ) Dϕ ( x ) + tDψ ( x ) | κ ( x, E s,t ) d H N − ( x ) dt Another observation is that the range of the ε for which this is true can be takento be the same for all s ∈ [ t , t ], since it depends on C bounds for the boundaries ∂ { ϕ ≥ s } (more precisely, on their largest curvature) and for ψ . Hence, if ε is smallenough, integrating (5.8) for s between t and t and using the co-area formula for BV functions, we obtain that J ( ˜ ϕ + ε ( ˜ ψ − ˜ ϕ )) − J ( ˜ ϕ ) ε = Z { t <ϕ 1] is a smooth, nondecreasing approximation of t + ( t − t ) χ { t ≥ t } . Passing to the limit, we obtain (5.3). (cid:3) Corollary 5.3. Let κ be a first variation of J in the sense of Definition 5.1, andassume it satisfies assumption C). Then, it is also the curvature in the sense ofDefinition 4.2.Proof. We need to prove that (4.5) and (4.6) hold with κ + = κ − = κ . To thispurpose, fix x ∈ ∂E and let { W n } ⊂ M with | W n ∩ E | > W n H → { x } .We can as before assume that E is the level set 1 / C ℓ,β function ϕ , constantout of a compact set, such that Dϕ = 0 in { ≤ ϕ ≤ } . Since J ( E ) = J ( { ϕ ≥ } ) + Z { / <ϕ< } κ ( x, E ϕ ( x ) ) dx and J ( E \ W n ) ≥ J ( { ϕ ≥ } ) + Z ( E \ W n ) \{ ϕ ≥ } κ ( x, E ϕ ( x ) ) dx, by (5.2) and (5.3) respectively, it follows that J ( E ) − J ( E \ W n ) ≤ Z W n ∩ E κ ( x, E ϕ ( x ) ) dx for n sufficiently large. Dividing both sides by | W n ∩ E | and letting n → ∞ , usingalso the continuity property of κ we conclude κ − ≤ κ . The opposite inequalityeasily follows by (5.1), choosing a sequence η n of smooth cut-off functions whosesupport concentrates around x and which are 1 in a neighborhood of x , definingΦ ε,n : y y − εη n ( y ) ν E ( x ), and setting W n := Φ ε n ,n ( E ) \ E , where ε n is chosenthrough a standard diagonal argument. The proof that κ + = κ is analogous. (cid:3) Examples of perimeters and their curvature In this part we present some examples of generalized perimeters and correspond-ing curvatures that fit into our theory. We will consider here, unless otherwisestated, that ℓ = 2, β = 0. The Euclidean perimeter. Let J be the Euclidean perimeter. More pre-cisely, let J be its lower semi-continuous extension to measurable sets introduced byCaccioppoli and De Giorgi. Then, J satisfies all the assumptions i)-vi). Moreover,let κ be the standard Euclidean curvature, i.e., the sum of the principal curvaturesof ∂E at x . It is standard that it is the first variation of the perimeter in the senseof Definition 5.1, hence by Propositions 5.2 we deduce that the Euclidean curvature κ is also the curvature of J in the sense of Definition 4.2.Clearly, the Euclidean perimeter is also uniformly continuous with respect to C inner variations of sets, namely it satisfies the continuity assumption C’). More ingeneral, any local curvature κ ( x, E ) that depends continuously on the normal andon the second fundamental form of E at x fits with our theory. For such local andposslbly anisotropic curvatures, we recover the well known existence and uniquenessof a viscosity solution to the geometric flows.6.2. The fractional mean curvature flow. Let α ∈ (0 , ), and let E ∈ M be such that χ E ∈ H α ( R N ). We recall that the α − fractional seminorm of thecharacteristic function of E is defined by[ χ E ] H α := (cid:18) (1 − α ) Z R N × R N | χ E ( x ) − χ E ( y ) || x − y | N +2 α dxdy (cid:19) . Let us introduce the generalized perimeter J ( E ) := [ χ E ] H α if χ E ∈ H α ( R N ) , + ∞ otherwise.Since the work in [4], this nonlocal perimeter has attracted much attention; werefer the interested reader to [30]. It is easy to check that J satisfies all the propertiesi)-vi), so that it fits with our notion of generalized perimeters.A notion of curvature corresponding to J has been introduced in [5], [22]: let ρ ( x ) := 1 / | x | N +2 α , ρ δ ( x ) = (1 − χ B (0 ,δ ) ( x )) ρ ( x ) . Then, for every E ∈ C set κ δ ( x, E ) = − − α ) Z R N ( χ E ( y ) − χ R N \ E ( y )) ρ δ ( x − y ) dy,κ ( x, E ) := lim δ → κ δ ( x, E ) . The curvature κ is well defined for all smooth sets, it is the first variation of theperimeter J and it is continuous with respect to C convergence. By Proposition 5.2we deduce that κ satisfies (5.2) and (5.3). In particular, κ is the curvature of J according with Definition 4.2.Indeed κ ( x, E ) is well defined for any set which satisfies an internal and externalball condition at x (see [22], [5]). In particular, κ is well defined for any E ∈ C , .This suggests that κ is a first order curvature. Let us show that this is the case. Proposition 6.1. Let Σ ∈ C , , let x ∈ ∂ Σ , and let ( p, X ) and ( p, Y ) be elementsof J , +Σ ( x ) and J , − Σ ( x ) , respectively. Then, (6.1) κ ∗ ( x, p, X, Σ) = κ ∗ ( x, p, Y, Σ) = κ ( x, Σ) . In particular, the curvature κ satisfies the first order curvature assumption (FO). ONLOCAL CURVATURE FLOWS 37 Proof. By Lemma 2.7 there exists a sequence ( p n , X n , E n ) → ( p, X, Σ) with uniformsuperjet at x , with E n ∈ C , such that(6.2) κ ( x, E n ) → κ ∗ ( x, p, X, Σ) . Moreover, we can always assume that E n → E in L (see Remark 2.9). SinceΣ ∈ C , , there exists r > B r ( x + r p | p | ) ⊂ Σ. Set˜ E n := E n ∪ B r ( x + r p n | p n | ) . Clearly, ( p n , X n , ˜ E n ) still converge to ( p, X, Σ) with uniform superjet at x . By thelower semicontinuity and monotonicity properties of κ we have(6.3) κ ∗ ( x, p, X, Σ) ≤ lim inf n κ ( x, ˜ E n ) ≤ lim inf n κ ( x, E n ) = κ ∗ ( x, p, X, Σ) . Moreover, since ˜ E n → Σ in L and ˜ E n satisfy a uniform internal and external ballcondition at x , it is easy to see (see for instance [22]) that κ ( x, ˜ E n ) → κ ( x, E ) , whichtogether with (6.3) proves that κ ∗ ( x, p, X, Σ) = κ ( x, Σ) . The proof for κ ∗ ( x, p, Y, Σ)is identical. (cid:3) Once proved that κ is a first order curvature, in view of Theorem 3.5 we recoverthe existence and uniqueness of a viscosity solution to the geometric flow, alreadyproved in [22]. Instead, the convergence of the corresponding minimizing movementscheme studied in Section 7 is completely new for this class of nonlocal perimetersand furnishes an approximation algorithm which is alternative to the threshold-dynamics-based one studied in [5]. Remark 6.2. It can be proved that in fact κ satisfies also the uniform continuityassumption C’). Thus, uniqueness could also be deduced from the second ordertheory of Subsection 3.2, but of course the ”first order” point of view is moreconvenient and straightforward in this case.We conclude this part giving a self contained proof of (5.2) and (5.3), which, inview of Proposition 5.3, yield that κ is the first variation of J .Let J δ ( E ) := (1 − α ) Z R N × R N | χ E ( x ) − χ E ( y ) | ρ δ ( x − y ) dxdy. We will first show that J δ , κ δ satisfy (5.2) and (5.3). Let W be a bounded measur-able set. Then,(6.4) Z W κ δ ( x, E ϕ ( x ) ) dx = − − α ) Z R N × R N χ W ( x )( χ E ϕ ( x ) ( y ) − χ R N \ E ϕ ( x ) ( y )) ρ δ ( x − y ) dydx = − (1 − α ) Z R N × R N ( χ W ( x ) − χ W ( y ))( χ E ϕ ( x ) ( y ) − χ R N \ E ϕ ( x ) ( y )) ρ δ ( x − y ) dydx ≤ Z R N × R N | χ W ( x ) − χ W ( y ) | ρ δ ( x − y ) dydx, with equality if and only if W = E s for some s ∈ (0 , δ → 0, the following limits hold:i) For every E ∈ M , J δ ( E ) → J ( E ), ii) Let ϕ : R N → (0 , 1) be such that E s := { ϕ ≥ s } are bounded and belongto C for every s ∈ (0 , 1) and D ϕ is negative definite on { ϕ = 1 } . Then, κ δ ( x, E ϕ ( x ) ) → κ ( x, E ϕ ( x ) ) in L loc ( R N ).This implies that also J , κ satisfies (5.3) and (5.2).6.3. General two body interaction perimeters. More in general one may con-sider a class of integral nonlocal perimeters of the form (see [22])(6.5) J K ( E ) := Z E Z R N \ E K ( x − y ) dxdy , where the (possibly singular) nonnegative kernel K satisfies:(i) K ∈ L ( R N \ B (0 , δ )) for all δ > r > e ∈ S N − we have that K ∈ L ( { z ∈ R N : r | z · e | ≤| z − ( z · e ) e | } ).The associated nonlocal curvature κ ( x, E ) := − Z R N ( χ E ( y ) − χ R N \ E ( y )) K ( x − y ) dy is well defined in the principal value sense provided that E satisfies both an innerand an outer ball condition at the point x ∈ ∂E . One can check that also these cur-vatures are covered by both the first order and second order theories of generalizedcurvatures.6.4. The flow generated by the regularized pre-Minkowski content. Let ρ > ρ -neighborhood of the boundary of E , i.e.,(6.6) M ρ ( E ) := | ( ∂E ) ρ | = | ( ∪ x ∈ ∂E B ρ ( x )) | . We refer to M ρ as the pre-Minkowski content of ∂E , since as ρ → | ( ∂E ) ρ | / ρ approximates the Minkowski content, which coincides with the standard perimeteron smooth sets.An issue with definition (6.6) is that it depends on the choice of the representa-tive within the Lebesgue equivalence class of the set E . For this reason, one mayintroduce the following variant:(6.7) J ρ ( E ) = 12 ρ Z R N osc B ( x,ρ ) ( χ E ) dx where osc A ( u ) denotes the essential oscillation of the measurable function u over ameasurable set A , defined by osc A ( u ) = ess sup A u − ess inf A u . One checks that J ρ ( E ) coincides with the measure of the ρ -neighborhood of the essential boundaryof E . Moreover, J ρ ( E ) = inf {M ρ ( E ′ ) : | E △ E ′ | = 0 } , where E △ E ′ denotes thesymmetric difference ( E \ E ′ ) ∪ ( E ′ \ E ).In [14] we have proved that the functional (6.7) is a generalized perimeter, wehave introduced the corresponding curvature, and studied the geometric flow. Letus introduce a notion of curvature corresponding to J ρ ; let E ∈ C , and denote by ν E ( x ) the outer normal unit vector to ∂E at x .For x ∈ ∂E , set(6.8) κ ρ ( x, E ) = κ outρ ( x, E ) + κ inρ ( x, E ) , ONLOCAL CURVATURE FLOWS 39 where(6.9) κ outρ ( x, E ) = ρ det( I + ρDν E ( x )) if dist( x + ρν E ( x ) , E ) = ρ , κ inρ ( x, E ) = − ρ det( I − ρDν E ( x )) if dist( x − ρν E ( x ) , E c ) = ρ , { < d E <ρ } (for κ outρ ) and {− ρ < d E < } (for κ inρ ) when the boundary is infinitesimallymodified at x , and their sum is a natural candidate for the curvature associated tothe energy E ρ . Indeed, in [14] we have proved that κ ρ ( x, E ) is the first variation of J ρ (in the classical sense (5.1)) whenever E ∈ C is such that the points at distance ρ from ∂E admit a unique projection on ∂E (indeed such condition can be weakeneda little). In order to have a well defined curvature for all E ∈ C , one can considerthe following regularization of J ρ : J f ( E ) = Z R N f ( d E ( x )) dx = Z ρ ( − sf ′ ( s )) J s ( E ) ds, where d E is the signed distance from ∂E and f : R → R + is even, smooth anddecreasing in R + , with support in [ − ρ, ρ ]. Such a regularization was consideredalso in [2] for numerical purposes.The corresponding curvature κ f is(6.11) κ f ( x, E ) = κ outf ( x, E ) + κ inf ( x, E ) , where κ outf ( x, E ) = Z ρ ( − sf ′ ( s )) κ outs ( x, E ) ds, κ inf ( x, E ) = Z ρ ( − sf ′ ( s )) κ ins ( x, E ) ds. Let r in be the maximal r ∈ [0 , ρ ] such that E satisfies the internal ball conditionwith radius r at x , and let r out be defined analogously. Clearly, r in , r out and thesecond fundamental form at x are uniformly continuous with respect to smoothinner variations. We immediately deduce that κ f satisfies the uniform continuityassumption C’).In [14] we have proved that κ f ( x, E ) is the curvature corresponding to J f , accord-ing to both Definitions 5.1 and 4.2, and we have studied the corresponding curvatureflow through the minimizing movements method. As a consequence of the analysisof this paper, namely by the Comparison Principle provided by Theorem 3.8, weget the new result that such a geometric evolution is indeed unique.6.5. The shape flow generated by p -capacity. In this subsection we show thatthe shape flow of bounded sets generated by the p -capacity fits into our generalframework. Notice that the case p = 2 yields an evolution that is similar to theHele-Show type flow considered in [8]. To this aim, given 1 < p < N , we consider the following relaxed p -capacity of aset E ∈ M defined by(6.12) Cap p ( E ) := inf (cid:26)Z R N | Dw | p dx : w ∈ K p and w ≥ E (cid:27) , where K p stands for the subspace of functions w of L p ∗ ( R N ) such that Dw ∈ L p ( R N ). Note that the above definition departs from the standard one in whichthe condition w ≥ E is replaced by E ⊂ ˚ { w ≥ } . It may be thought asa sort of L -lower semicontinuous envelope of the standard p -capacity, having theproperty of being insensitive to negligible sets and thus independent of the Lebesguerepresentative of E . Clearly the two definitions coincide on open sets and it is notdifficult to check that they also agree on all closed sets F such that F = ˚ F , with | ∂F | = 0, in particular on all sets in C .Formula (6.12) does not provide yet a generalized perimeter. Indeed, Cap p ( R N ) =+ ∞ and, more in general, if E ∈ C , then Cap p ( E ) < + ∞ if and only if Cap p ( R N \ E ) = + ∞ . Thus, the requirements i) and ii) stated at the begining of Subesction 4.1are not fulfilled. On the other hand, properties iii) and vi) are evident, the lower-semicontinuity iv) follows in a standard way, while the submodularity property v)can be proven as in the case of the standard capacity (see [19, Theorem 2-(vii) ofSection 4.7]). Since our focus will be on the evolution of bounded sets, we will builda generalized perimeter J p , by enforcing the following properties:a) J p ( E ) = Cap p ( E ) for all bounded sets E ∈ M ;b) J p ( E ) = J p ( R N \ E ) for all E ∈ M .This is achieved by setting(6.13) J p ( E ) := min { Cap p ( E ) , Cap p ( R N \ E ) } for all E ∈ M . It follows immediately from the definition and from the properties of Cap p ( · ) recalled above that J p satisfies i)–iv) of Subsection 4.1 and the translationinvariance vi). It only remains to check the submodularity property v). To thisaim, let us consider the case of two sets E , F ∈ M such that Cap p ( E ) < + ∞ and Cap p ( R N \ F ) < + ∞ . As Cap p ( R N \ E ) = Cap p ( F ) = + ∞ , we have J p ( E ) = Cap p ( E ) and J p ( F ) = Cap p ( R N \ F ). Moreover, since Cap p ( E ∩ F ) ≤ Cap p ( E ), wealso have J p ( E ∩ F ) = Cap p ( E ∩ F ), while the fact that Cap p ( E ∪ F ) ≥ Cap p ( F ) =+ ∞ implies J p ( E ∪ F ) = Cap p ( R N \ ( E ∪ F )). Thus, in this case the submodularityinequality is equivalent to Cap p ( R N \ ( E ∪ F )) + Cap p ( E ∩ F ) ≤ Cap p ( E ) + Cap p ( R N \ F ) , which is obviously true since Cap p ( R N \ ( E ∪ F )) ≤ Cap p ( R N \ F ) and Cap p ( E ∩ F ) ≤ Cap p ( E ) by the non-decreasing monotonicity of the set function Cap p ( · ). Since allthe remaining cases are either trivial or reduce the the submodularity of Cap p ( · ),also property v) is established for J p , which is therefore a generalized perimeter.By a standard application of the Direct Method of the Calculus of Variations onemay also check the existence of a unique capacitary potential w E associated with anyset E , i.e., of a unique solution to the problem (6.12), whenever Cap p ( E ) < + ∞ .The Euler-Lagrange conditions for (6.12) easily yield that w E is super p -harmonic ONLOCAL CURVATURE FLOWS 41 in R N , in fact it is determined as the unique solution w E ∈ K p to(6.14) − Z R N | Dw E | p − Dw E Dϕ dx ≥ ϕ ∈ K p , with ϕ ≥ E . w E = 1 a.e. in E .Denoting by E (0) the set of points with vanishing density with respect to E , itfollows in particular that w E is p-harmonic in the interior of E (0) .In order to identify the nonlocal curvature corresponding to J p ( · ), we exploitthe theory developed in Section 5. Let E ∈ C and bounded, and let (Φ ε ) ε be aone-parameter family of diffeomorphisms from R N onto itself of class C both in ε and x and such that Φ ( x ) = x for all x ∈ R N . Denote ψ ( x ) := ∂ Φ ε ( x ) ∂ε | ε =0 . Thenby the Hadamard formulae (see for instance [29]) one has(6.15) dd ε J p (Φ ε ( E )) | ε =0 = dd ε Cap p (Φ ε ( E )) | ε =0 = dd ε Z R N \ Φ ε ( E ) | Dw Φ ε ( E ) | p dx (cid:12)(cid:12) ε =0 = Z ∂E | Dw E | p ( x ) ψ ( x ) · ν E ( x ) d H N − ( x ) , where, as usual, ν E denotes the outer unit normal to E . Motivated by the aboveformula and recalling that J p ( E ) = Cap p ( R N \ E ) for E ∈ C and unbounded, forevery E ∈ C and x ∈ ∂E we set(6.16) κ p ( x, E ) := | Dw E ( x ) | p = (cid:12)(cid:12)(cid:12)(cid:12) ∂w E ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p if E is bounded, −| Dw R N \ E ( x ) | p = − (cid:12)(cid:12)(cid:12)(cid:12) ∂w R N \ E ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p if E is unbounded.Recalling that w E is p-harmonic on E c and satisfies the Dirichlet condition w E = 1on ∂E , the well-established regularity theory for the p -Laplacian (see for instance[24]) yields that w E is of class C ,α up to the boundary for all α ∈ (0 , C ,α -norm of Dw E depending only on its L p -norm and the C ,α -norm of ∂E . Infact, whenever E n → E in C ,α and x ∈ ∂E ∩ ∂E n we have(6.17) ∂w E n ∂ν ( x ) → ∂w E ∂ν ( x )as n → ∞ . In particular, it follows that the nonlocal curvature κ p defined in (6.16)satisfies the continuity property C) of Subsection 2.1.In turn, by Corollary 5.3 the set function (6.16) is the curvature associated with J p in the sense of Definition 4.2. Lemma 4.3 now implies that the monotonicityproperty A) stated in Subsection 2.1 holds for κ p .Since the translation invariance of κ p is evident, we have shown that (6.16)satisfies axioms A), B), and C) of Subsection 2.1. We recall that these axiomsare enough to guarantee the convergence (up to subsequences) of the minimizingmovements scheme studied in Section 7 to a viscosity solution of the correspondinglevel set equation.It remains to investigate the uniqueness. Instead of establishing the reinforcedcontinuity property C’), we check that the nonlocal perimeter J p generates a “first-order” flow and we apply the theory of Subsection 3.1. To this aim, denote by ( κ p ) ∗ and ( κ p ) ∗ the lower and the upper semicontinuous extensions of κ p provided by formulas (2.9) and (2.10), respectively. Note that, as a straightforward consequenceof the definition and of (2.10), we have(6.18) ( κ p ) ∗ ( x, p, X, E ) = − ( κ p ) ∗ ( x, − p, − X, R N \ E )for E ∈ M , x ∈ ∂E , and ( p, X ) ∈ J , + E ( x ).We are now in a position to prove that condition (FO) of Subsection 3.1 is sat-isfied. Uniqueness will then follow by applying the Comparison Principle providedby Theorem 3.5. Lemma 6.3. Let Σ ⊂ R N belong to C , . Let x ∈ ∂ Σ and let ( p, X ) and ( p, Y ) beelements of J , +Σ ( x ) and J , − Σ ( x ) , respectively. Then, ( κ p ) ∗ ( x, p, X, Σ) = ( κ p ) ∗ ( x, p, Y, Σ) . Proof. In light of (6.18), it is enough to consider the case of a bounded set Σ ofclass C , . Let w Σ be the associated capacitary potential. The conclusion of thelemma will be achieved by showing that(6.19) ( κ p ) ∗ ( x, p, X, Σ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p = ( κ p ) ∗ ( x, p, Y, Σ) . To this aim, let E ⊇ Σ be a bounded set of C admissible for the Definition 2.9 of( κ p ) ∗ ( x, p, X, Σ), and let w E be the corresponding capacitary potential. Recall thatby (6.14), we have that w E is super p-harmonic in R N \ Σ, while w Σ is p-harmonicin the same set. Since w E = w Σ = 1 on ∂ Σ, by the Maximum Principle we inferthat 1 ≥ w E ≥ w Σ in R N \ Σ. In turn, κ p ( x, E ) = (cid:12)(cid:12)(cid:12)(cid:12) ∂w E ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p and therefore we may conclude that(6.20) ( κ p ) ∗ ( x, p, X, Σ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p . To show the opposite inequality, fix δ > n ) n ⊂ C , with the following properties:i) Σ ⊂ ˚Σ n ∪ { x } , with ∂ Σ n ∩ ∂ Σ = { x } ;ii) ( p, X + δI ) ∈ J n ( x ) for all n ∈ N ;iii) Σ n → Σ in the C ,α -sense, for all α ∈ (0 , n -level sets of the signed distance function from Σ and modify them inthe proximity to x in order to fulfill conditions i) and ii). By (6.17), for any givensmall ε > n such that(6.21) (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ¯ n ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p − ε . Recall now that from the proof of Lemma 2.7, we may construct a decreasingsequence of sets ( E n ) n ⊂ C such thata) E n ց Σ in the Hausdorff sense;b) ( p, X + δ n I ) ∈ J E n ( x ) for some δ n ց κ p ( x, E n ) = (cid:12)(cid:12)(cid:12) ∂w En ∂ν ( x ) (cid:12)(cid:12)(cid:12) p → ( κ p ) ∗ ( x, p, X, Σ). ONLOCAL CURVATURE FLOWS 43 Taking into account i) and ii) above, it follows from a) and b) that E n ⊂ Σ ¯ n for n large enough. For all such n ’s, by the Maximum Principle as in the first part of theproof, we have (cid:12)(cid:12)(cid:12)(cid:12) ∂w E n ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ¯ n ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p − ε , where in the last inequality we have used (6.21). By c), passing to the limit in theleft-hand side of the above formula and by the arbitrariness of ε , we deduce( κ p ) ∗ ( x, p, X, Σ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂w Σ ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p , which, together with (6.20), establishes the first equality in (6.19). The secondequality can be proven in a completely analogous fashion. (cid:3) The minimizing movements approximation In this section we implement the minimizing movements scheme to solve andapproximate the nonlocal κ -curvature flow, in the spirit of [1, 25]. We extend theapproach of [12] (see also [14]) to our general framework.7.1. The time-discrete scheme for bounded sets. We start by introducing theincremental minimum problem. To this purpose, given a bounded set E = ∅ , we let(7.1) d E ( x ) = dist( x, E ) − dist( x, R N \ E )be the signed distance function to ∂E . Fix a time step h > (cid:26) J ( F ) + 1 h Z F d E ( x ) dx : F ∈ M (cid:27) . Note that Z F d E ( x ) dx − Z E d E ( x ) dx = Z E ∆ F dist( x, ∂E ) dx so that (7.2) is equivalent tomin (cid:26) J ( F ) + 1 h Z E ∆ F dist( x, ∂E ) dx : F ∈ M (cid:27) . Proposition 7.1. The problem (7.2) admits a minimal and a maximal solution.Proof. Since the functional J extended to L loc according to (4.3) is convex, it iseasy to check that the minimization problem(7.3) min u ∈ L ∞ ( R N ;[0 , J ( u ) + 1 h Z R N u ( x ) d E ( x ) dx admits a solution. Then, observe that(7.4) J ( u ) + 1 h Z R N u ( x ) d E ( x ) dx = Z J ( { u > s } ) + 1 h Z { u>s } d E ( x ) dx ! ds , from which we easily deduces that for a.e. s ∈ [0 , { u > s } is a solution to (7.2).Let now E and E be two solutions to (7.2). Then again by (7.4) their characteristicfunctions and in turn, by convexity, ( χ E + χ E ) are solutions to (7.3). Sincealmost all their superlevel sets are solutions to (7.2), we deduce, in particular, that E ∩ E and E ∪ E are solutions to (7.2). Finally let E n be a sequence of solutionsto (7.2) such that | E n | → m := inf {| E | : E is a solution to (7.2) } . Then, F k := ∩ kn =1 E n is a decreasing sequence of solutions such that | F k | → m .Thus, by semicontinuity, their L -limit E := ∩ ∞ n =1 E n is the minimal solution. Theexistence of a maximal solution can be proven analogously. (cid:3) For any bounded set E = ∅ we let T + h E and T − h E denote the maximal and theminimal solution of (7.2), respectively. We also set T ± h ∅ := ∅ . We will mainly useminimal solutions, and write T h E := T − h E . This choice corresponds to consideropen superlevels in our level set approach (see Proposition 7.12). It is convenientto fix a precise representative for T ± h E . To this purpose, we will identify anymeasurable set with the representative given by the set of Lebesgue points of thecharacteristic function. Lemma 7.2. If E ⊆ E ′ , then T ± h E ⊆ T ± h E ′ .Proof. The proof is classical and we just sketch it. We first assume that E ⊂⊂ E ′ ,so that d E > d E ′ everywhere. We compare the energy (7.2) of T ± h E with the one of T ± h E ∩ T ± h E ′ , and the energy (7.2) (with E replaced by E ′ ) of T ± h E ′ with the one of T ± h E ∪ T ± h E ′ . We sum both inequalities and use (4.1) to deduce that T ± h E ⊆ T ± h E ′ .Now we conclude the proof by a perturbation argument. For ε > F ε be theminimal solution of (7.2) with d E replaced by d E + ε . Arguing as before, we deducethat F ε are increasing in ε and F ε ⊆ T − h E ′ . Therefore F ε → F := ∪ ε F ε in L loc . Bylower semicontinuity it follows that F is a solution, and thus T − h E ⊆ F ⊆ T − h E ′ .The inclusion T + h E ⊆ T + h E ′ can be proven similarly. (cid:3) Remark 7.3. Let f be a measurable function such that f − := − f ∧ ∈ L ( R N ).Then one can argue as in Proposition 7.1 to prove that the minimum problemmin (cid:26) J ( F ) + Z F f dx : F ∈ M (cid:27) admits a minimal and a maximal solution, denoted by E − f and E + f respectively.Moreover, arguing exactly as in the proof of Lemma 7.2, one can show that if f , f are measurable functions with f − , f − ∈ L ( R N ) and f ≤ f a.e., then E ± f ⊆ E ± f . Lemma 7.4. If E + B R ⊆ E ′ , then ( T ± h E ) + B R ⊆ T ± h E ′ .Proof. By Lemma 7.2 for every z ∈ B R we have T ± h ( E + z ) ⊆ T ± h E ′ . By translationinvariance we conclude( T ± h E ) + B R = [ z ∈ B R ( T ± h E ) + z = [ z ∈ B R T ± h ( E + z ) ⊆ T ± h E ′ . (cid:3) Lemma 7.5. For any R > we have T ± h ( B R ) ⊆ B CR , where C depends only onthe dimension N . ONLOCAL CURVATURE FLOWS 45 Proof. By Lemma 7.4 we have(7.5) T ± h ( B R ) + B R ⊆ T ± h ( B R ) . Let c > 1, and assume there exists x ∈ T ± h ( B R ) \ B cR . Since in particular x ∈ T ± h ( B R ), by (7.5) we have B ( x, R ) ⊆ T ± h ( B R ). Hence0 > J ( T ± h ( B R )) + 1 h Z T ± h ( B R ) | y | − R dy ≥ h Z B ( x,R ) | y | − R dy + Z T ± h ( B R ) \ B ( x,R ) | y | − R dy ! ≥ h Z B ( x,R ) | y | − R dy + Z B R | y | − R dy ! ≥ h (cid:18)Z B R ( c − R − | y | dy + Z B R | y | − R dy (cid:19) which is positive if c is large enough (depending only on the dimension), a contra-diction. (cid:3) Next lemma provides a more refined estimate. Lemma 7.6. Let C > be such that the statement of Lemma 7.5 holds, and let c , c be as in (2.2) . Then, the following holds. i) Let R > . Then, for every h > such that R − hc ( CR ) > we have T ± h B R ⊆ B R − hc ( CR ) . ii) Let R > and σ > be fixed. Then, for h > small enough (dependingon R and σ ), we have T ± h B R ⊇ B R − hc ( R/σ ) for all R ≥ R .Proof. First, we know from the previous result that T ± h B R ⊆ B CR . Proof of i). We can always assume T ± h B R = ∅ . Let ¯ ρ = sup { ρ ∈ [0 , CR ] : | T ± h B R \ B ρ | > } . Let ¯ x ∈ ∂B ¯ ρ such that | T ± h B R ∩ B (¯ x, ε ) | > ε > 0, andlet ρ > ¯ ρ . Let τ ∈ R N be such that B ( − τ, ρ ) ⊃ B ¯ ρ and ∂B ( − τ, ρ ) is tangent to ∂B ¯ ρ at ¯ x ; i.e., τ = ( ρ/ ¯ ρ − x .We let for ε > B ε = B ( − (1 + ε ) τ, ρ ) and W ε = T ± h B R \ B ε . Notice thatby construction W ε has positive measure and converges to ¯ x in the Hausdorff senseas ε → 0. By submodularity we have(7.6) J ( B ε ∩ T ± h B R ) + J ( B ε ∪ T ± h B R ) ≤ J ( B ε ) + J ( T ± h B R ) . By (7.6) and using the minimality of T ± h B R we have J ( B ( − τ, ρ ) ∪ ( W ε + ετ )) − J ( B ( − τ, ρ )) = J ( B ε ∪ W ε ) − J ( B ε ) ≤ J ( T ± h B R ) − J ( B ε ∩ T ± h B R ) ≤ − h Z W ε | x | − R dx . Dividing the previous inequality by | W ε | and passing to the limit as ε → 0, in viewof the very definition (4.5) of κ we get κ ( B ( − τ, ρ )) ≤ h ( R − | ¯ x | ) = 1 h ( R − ¯ ρ ) . Recalling the definition of c and the fact that it is a continuous decreasing function,we deduce the thesis by sending ρ → ¯ ρ . Proof of ii). Assume x is such that¯ ρ = max { ρ > | B ( x , ρ ) \ T ± h B R | = 0 } ∈ ]0 , CR ] . As in the proof of i), we can find ¯ x ∈ ∂B ( x , ¯ ρ ) such that | B (¯ x, ε ) \ T ± h B R | > ε > 0, we fix ρ < ¯ ρ and set τ = (1 − ρ/ ¯ ρ )(¯ x − x ), so that { ¯ x } = ∂B ( x , ¯ ρ ) ∩ ∂B ( x + τ, ρ ). We let B ε = B ( x + (1 + ε ) τ, ρ ) and define W ε = B ε \ T ± h B R . Bysubmodularity we have J ( B ε ∩ T ± h B R ) + J ( B ε ∪ T ± h B R ) ≤ J ( B ε ) + J ( T ± h B R ) . Using the minimality of T ± h ( B R ) we deduce J ( B ε \ W ε ) − J ( B ε ) ≤ J ( T ± h B R ) − J ( B ε ∪ T ± h B R ) ≤ h Z W ε | x | − R dx. Dividing the previous inequality by | W ε | and passing to the limit as ε → 0, in viewof the very definition of κ (4.6) we get − κ (¯ x, B ε ) ≤ h ( | ¯ x | − R ) . It follows that | ¯ x | ≥ R − hc (¯ ρ ).Now, let C be the constant of Lemma 7.5, and choose h so small that J ( B R / C ) + 1 h Z B R / C | x | − R C dx ≤ J ( B R / C ) − R Ch | B R / (8 C ) | < , so that T ± h B R/ C = ∅ . Note that B R/ C + B R/ ⊆ B R . Thus, by Lemma 7.4 T ± h ( B R/ C ) + B R/ ⊆ T ± h B R . In particular, if x ∈ T ± h B R/ (4 C ) it follows that B ( x , R ) ⊆ T ± h B R . By the first part of the proof of ii), we find that B ( x , | ¯ x − x | ) ⊆ T ± h B R for some ¯ x with | ¯ x | ≥ R − hc (3 R/ x ∈ T ± h B R/ (4 C ) ⊆ B R/ , we obtain that B R/ ⊆ T ± h B R ,provided that h is small enough. We can now use again the previous analysis with x = 0, ¯ ρ ≥ R/ h is small enough, B R − hc ( R/ ⊆ T ± h B R .Applying once again the first part of the proof with x = 0 and ¯ ρ ≥ R − hc ( R/ 4) weconclude that, if h is small enough, B R − hc ( R/σ ) ⊆ B R − hc ( R − hc ( R/ ⊆ T ± h B R . (cid:3) The time discrete scheme for unbounded sets. Here we show how toextend the time discrete scheme to the case of unbounded sets with bounded com-plement. To this purpose, we introduce the perimeter ˜ J defined as˜ J ( E ) := J ( R N \ E ) for all E ∈ M . Note that ˜ J satisfies all the structural assumptions of generalized perimeters. Let ˜ κ be the corresponding curvature. Then, it is easy to see that ˜ κ ( x, E ) = − κ ( x, R N \ E ),and thus max x ∈ ∂B ρ max { ˜ κ ( x, B ρ ) , − ˜ κ ( x, R N \ B ρ ) } = c ( ρ )min x ∈ ∂B ρ min { ˜ κ ( x, B ρ ) , − ˜ κ ( x, R N \ B ρ ) } = c ( ρ ) , (7.7)where c ( ρ ) , c ( ρ ) are the functions defined in (2.1) and (2.2). ONLOCAL CURVATURE FLOWS 47 For every bounded set F we denote by ˜ T ± h ( F ) the maximal and the minimal so-lution to problem (7.2), according to Proposition 7.1 with J replaced by ˜ J . Finally,for every E ⊆ R N such that F := R N \ E is bounded we set(7.8) T ± h E := R N \ ˜ T ∓ h ( R N \ E ) . As in the case of bounded sets, we let T h E := T − h E .Taking into account also (7.7), one can easily check that Lemmas 7.5 and 7.6translate into the following statements: Lemma 7.7. For any R > we have R N \ B CR ⊆ T ± h ( R N \ B R ) , where C dependsonly on the dimension N . Lemma 7.8. Let C > be such that the statement of Lemma 7.5 holds, and let c , c be as in (2.2) . Then, the following holds: i) Let R > . Then, R N \ B R − hc ( CR ) ⊆ T ± h ( R N \ B R ) for every h > suchthat R − hc ( CR ) > ; ii) Let R > and σ > be fixed. Then, for h > small enough (dependingon R and σ ), we have T ± h ( R N \ B R ) ⊆ R N \ B R − hc ( R/σ ) for all R ≥ R . Remark 7.9. A consequence of Lemmas 7.5, 7.7 is that T h B R ⊆ B R + hK and R N \ B R + hK ⊆ T h ( R N \ B R ) for any h > R > 0, where K is defined in(2.3). In particular, iterating these estimates, we deduce that T [ t/h ] h B R ⊆ B R + tK and R N \ B R + tK ⊆ T [ t/h ] h ( R N \ B R ). In the limit as h → 0, we will get an estimatefor the extinction time of balls in the superlevels of our level set function (seeProposition 7.15).Note now that by Lemma 7.2 (applied to ˜ J in place of J ) and (7.8) if E , E areunbounded sets with compact boundary, then E ⊆ E = ⇒ T ± h E ⊆ T ± h E . It remains to consider the case of E bounded and E unbounded. Lemma 7.10. Let E ∈ M be bounded and let E ∈ M be unbounded, with compactboundary, and such that E ⊆ E . Then, T ± h E ⊆ T ± h E .Proof. Choose R > E , R N \ E ⊆ B R and note that by Lemmas 7.2and 7.5 (applied to ˜ J in place of J ) we get(7.9) R N \ T + h E = ˜ T − ( R N \ E ) ⊆ ˜ T − B R ⊆ B CR for some C > N . Recall that ˜ T − h ( R N \ E ) is the minimalsolution of min (cid:26) J ( R N \ F ) + 1 h Z F d R N \ E dx : F ∈ M (cid:27) . Considering the change of variable e F := R N \ F and using that d R N \ E = − d E , weeasily infer that T + h E = R N \ ˜ T − h ( R N \ E ) is the maximal solution of min (cid:26) J ( e F ) − h Z R N \ e F d E dx : e F ∈ M (cid:27) = min (cid:26) J ( e F ) + 1 h Z B CR d E dx − h Z R N \ e F d E dx : e F ∈ M (cid:27) − h Z B CR d E dx . Note now that Z e F d E χ B CR dx = Z B CR d E dx − Z R N \ e F d E dx for every e F with R N \ e F ⊆ B CR . It follows, also by (7.9), that T + h E is the maximalsolution of(7.10) min (cid:26) J ( e F ) + 1 h Z e F d E χ B CR dx : e F ∈ M , R N \ e F ⊆ B CR (cid:27) . By the same reasoning, one can show that T − h E is the minimal solution of (7.10).Observing that d E χ B CR ≤ d E and that T ± h E ∪ T ± h E , T ± h E ∩ T ± h E are admissiblecompetitors for (7.10), one can argue exactly as in the proof of Lemma 7.2 toconclude that T ± h E ⊆ T ± h E . (cid:3) The level-set approach. Given any bounded uniformly continuous function u : R N → R , constant outside a compact set, we introduce a transformation of u which is defined by applying T h to all the superlevel sets of u . This is standard andhas been done in a similar geometric setting in many papers (see [11, 17]).To this purpose, notice that all the superlevels of u are either bounded or withbounded complement, and that for any couple of levels s > s ′ ∈ R we have { u >s } ⊆ { u > s ′ } . Thus, in view of Lemma 7.2 we have T h { u > s } ⊆ T h { u > s ′ } .Let ω : R + → R + an increasing, continuous modulus of continuity for u . Since { u > s } + B ω − ( s − s ′ ) ⊆ { u > s ′ } , by Lemma 7.4 we deduce that T h { u > s } + B ω − ( s − s ′ ) ⊆ T h { u > s ′ } . It follows that the sets T h { u > s } are themselves the level sets { v > s } of a uniformlycontinuous function v =: T h u , with the same modulus of continuity. More precisely,we set T h u ( x ) := sup { λ ∈ R : x ∈ T h { u > λ }} . Notice that, by Lemmas 7.5 and 7.7,also T h u is constant out of a compact set. Moreover, if u ≥ u ′ , then T h u ≥ T h u ′ .In the two following propositions, equality between sets must be understood up tonegligible sets. Proposition 7.11. For every λ ∈ R we have T h ( { u > λ } )) = T − h ( { u > λ } )) = { T h u > λ } . Analogously, T + h ( { u ≥ λ } )) = { T h u ≥ λ } . Proof. For every δ ≥ E δ := T h ( { u > λ + δ } ) , A δ := { T h u > λ + δ } . ONLOCAL CURVATURE FLOWS 49 We have to prove that E = A . First, notice that by the very definition of T h u ,for every δ ≥ A δ ⊆ E δ so that in particular A ⊆ E . To prove the reverse inclusion, observe that(7.12) d { u>λ + δ } → d { u>λ } uniformly as δ → 0. Moreover, A δ ր A in L loc as δ → J , it easily follows that A is a solution of (7.2) with E replaced by { u > λ } (or of (7.10) in the unbounded case, with E replaced by { u > λ } ) . Moreover, by (7.11) it is the minimal one, i.e., it coincides with E . Thesimilar proof of the second statement is left to the interested reader. (cid:3) Given a continuous function u constant outside of a bounded set, define(7.13) u h ( x, t ) = T [ t/h ] h ( u )( x )for every h > t ≥ 0, where [ · ] denotes the integer part.Proposition 7.11 applied to u h ( · , ( k − h ) yields the following: Proposition 7.12. For every h, k > and for every λ ∈ R we have T − h ( { u h ( · , ( k − h ) > λ } ) = { u h ( · , kh ) > λ } and T + h ( { u h ( · , ( k − h ) ≥ λ } ) = { u h ( · , kh ) ≥ λ } . We have seen that for all t , u h ( · , t ) is uniformly continuous (with the same mod-ulus ω as u ). Let us now study the regularity in time of this function. Lemma 7.13. For any ε > , there exists τ > and h > (depending on ε ) suchthat for all | t − t ′ | ≤ τ and h ≤ h we have | u h ( · , t ) − u h ( · , t ′ ) | < ε .Proof. Let ε > R := ω − ( ε/ / 2. Since ω is a modulus of continuity for u h it readly follows that for every x (7.14) B ( x, ω − ( ε/ ⊆ { u h ( · , t ) > u h ( x, t ) − ε } . We only treat the case where { u h ( · , t ) > u h ( x, t ) − ε } is bounded, the other beinganalogous. Let τ := R /c ( R / c is amonotone decreasing function, there exists h depending on R such that(7.15) B ( x, R ) ⊆ B ( x, ω − ( ε/ − nhc ( R / ⊆ T nh B ( x, ω − ( ε/ ω − ( ε/ − nhc ( R / ≥ R , i.e., as long as nh ≤ τ .Now, let t ′ > t such that t ′ − t ≤ τ , and let n := [( t ′ − t ) /h ]. Since nh ≤ τ , by(7.14), (7.15), Lemma 7.2, and Proposition 7.12 we have { u h ( · , t ′ ) > u h ( x, t ′ ) − ε } = { u h ( · , t + nh ) > u h ( t, x ) − ε } = T nh { u h ( · , t ) > u h ( x, t ) − ε } ⊇ T nh B ( x, ω − ( ε/ ⊇ B ( x, R ) . In particular, u h ( x, t ′ ) > u h ( x, t ) − ε. In order to show u h ( x, t ′ ) < u h ( x, t ) + ε we proceed in a similar way. Precisely, we observe that B ( x, ω − ( ε/ ⊆ { u h ( · , t ) < u h ( x, t ) + ε } , that is, { u h ( · , t ) ≥ u h ( x, t ) + ε } ⊆ R N \ B ( x, ω − ( ε/ . We then proceed as in the first part of the proof, but now using Lemma 7.7 insteadof Lemma 7.5 and Lemma 7.10 instead of Lemma 7.2. (cid:3) Convergence analysis. In this subsection we show that any limit of thediscrete evolutions is a viscosity solution. Recalling Lemma 7.13 and the uniformcontinuity in space of u h , by a straightforward variant of Ascoli-Arzel`a’s Theoremwe deduce the precompactness of u h . Moreover, in view of Remark 7.9 we deducealso that the limit u is constant out of a compact set. Summarizing, the followingproposition holds. Proposition 7.14. Let T > . Up to a subsequence, u h converges uniformly on R N × [0 , T ] as h → to a function u ( x, t ) , which is bounded and uniformly contin-uous, and constant out of a compact set. For every r > 0, set(7.16) ˆ c ( r ) := max { , c ( r ) } . Given r > 0, let r ( t ) be the solution of the following ODE(7.17) ( ˙ r ( t ) = − ˆ c ( r ( t )); r (0) = r Notice that (7.17) admits a unique solution r ( t ) until some extinction time T ∗ ( r )with r ( T ∗ ) = 0. Proposition 7.15. Let u ( x, t ) be the function given by Proposition 7.14, let λ ∈ R ,and let B ( x , r ) ⊂ { u ( · , t ) > λ } . Then, B ( x , r ( t − t )) ⊂ { u ( · , t ) > λ } for every t ≤ T ∗ ( r ) + t , where r ( t ) is the solution of the ODE (7.17) and T ∗ ( r ) is itsextinction time. The same statement holds by replacing the superlevel of u with itssublevel.Proof. We only treat the case of { u ( · , t ) > λ } bounded, since the other one isanalogous. By assumption, if R < r , for h small enough B ( x , R ) ⊂ { u h ( · , t ) >λ } . Let σ > R be defined recursively by R n +1 = R n − hc ( R n /σ ). ByLemmas 7.2, 7.6, and 7.12 one has that B ( x , R [( t − t ) /h ]+1 ) ⊂ { u h ( · , t ) > λ } for t ≥ t , as long as R [( t − t ) /h ]+1 > 0. Let also r σ be the unique solution of ˙ r σ ( t ) = − ˆ c ( r σ ( t ) /σ ) with initial value r σ (0) = R . One observes that if r σ ( nh ) ≤ R n , then r σ (( n + 1) h ) ≤ R n − Z ( n +1) hnh ˆ c (cid:18) r σ ( s ) σ (cid:19) ds ≤ R n − Z ( n +1) hnh ˆ c (cid:18) R n σ (cid:19) ds ≤ R n − Z ( n +1) hnh c (cid:18) R n σ (cid:19) ds = R n +1 since ˆ c is nondecreasing. As a consequence, B ( x , r σ ( h [( t − t ) /h ] + h ) ⊂ { u h ( · , t ) >λ } for t ≥ t as long as the radius is positive. We conclude sending h → 0, then ONLOCAL CURVATURE FLOWS 51 R → r and σ → 1. The proof of the last part of the proposition is very similar.One observes that by Lemmas 7.10, 7.8, and 7.12, we have (with the same definitionof R n ) { u h ( · , t ) > λ } ⊂ R N \ B ( x , R [( t − t ) /h ]+1 ), that is B ( x , R [( t − t ) /h ]+1 ) ⊂{ u h ( · , t ) ≤ λ } for t ≥ t , as long as R [( t − t ) /h ]+1 > 0. The conclusion then followsas before. (cid:3) We are now in a position to state and prove the main result of this section. Theorem 7.16. The function u provided by Proposition 7.14 is a viscosity solutionof the Cauchy problem (2.4) in the sense of Definition 2.10. Remark 7.17. We observe that this holds under assumptions C) and D) on thecurvature. If in addition C’) holds, then the limit flow is unique and one also deducesthat the whole family ( u h ) h> converges uniformly as h → Proof. We denote by u h k a subsequence of u h converging to u . Let us prove that u is a subsolution (the proof that it is a supersolution is identical). Let (¯ x, ¯ t ) ∈ R N × (0 , T ). Let ϕ be a C ℓ,β admissible test function at (¯ x, ¯ t ), and assume that(¯ x, ¯ t ) is a maximum point of u − ϕ . We need to show that(7.18) ∂ϕ∂t (¯ x, ¯ t ) + | Dϕ (¯ x, ¯ t ) | κ ∗ (¯ x, Dϕ (¯ x, ¯ t ) , D ϕ (¯ x, ¯ t ) , { ϕ ( · , ¯ t ) ≥ ϕ (¯ x, ¯ t ) } ) ≤ . Step 1. Let us first assume that Dϕ (¯ x, ¯ t ) = 0. By Remark 2.12 we can assumethat this is a strict maximum point and that ϕ is smooth.If the maximum is strict, then by standard methods we can find ( x k , t k ) → (¯ x, ¯ t ) such that u h k − ϕ has a maximum at ( x k , t k ). Moreover, for k large enough, Dϕ ( x k , t k ) = 0. We have that for all ( x, t ),(7.19) u h k ( x, t ) ≤ ϕ ( x, t ) + c k where c k := [ u h k ( x k , t k ) − ϕ ( x k , t k )], with equality if ( x, t ) = ( x k , t k ).Let η > ϕ ηh k ( x ) = ϕ ( x, t k ) + c k + η Q ( x − x k ) , where Q is as in Lemma 2.11 and Q ( z ) = | z | for | z | sufficiently small. Then, forall x ∈ R N , u h k ( x, t k ) ≤ ϕ ηh k ( x )with equality if and only if x = x k . We set l k := u h k ( x k , t k ) = ϕ ηh k ( x k ).By Lemma 2.11, we can assume that η is such that the superlevel sets { ϕ ηh k ≥ l k } are not critical for all k . Let ε > W ε := { x ∈ R N : u h k ( x, t k ) ≥ l k − ε } \ { x ∈ R N : ϕ ηh k ( x ) ≥ l k } . It is easy to see that for ε > | W ε | > 0, and converges to { x k } inthe Hausdorff sense as ε → 0. Now, if { u h k ( · , t k ) ≥ l k − ε } is bounded, by minimality we have(7.21) J ( { u h k ( · , t k ) ≥ l k − ε } ) + 1 h k Z { u hk ( · ,t k ) ≥ l k − ε } d { u hk ( · ,t k − h k ) ≥ l k − ε } ( x ) dx ≤ J ( { u h k ( · , t k ) ≥ l k − ε } ∩ { ϕ ηh k ≥ l k } )+ 1 h k Z { u hk ( · ,t k ) ≥ l k − ε }∩{ ϕ ηhk ≥ l k } d { u hk ( · ,t k − h k ) ≥ l k − ε } ( x ) dx . Adding to both sides the term J ( { u h k ( · , t k ) ≥ l k − ε } ∪ { ϕ ηh k ≥ l k } ) and using (4.1),we obtain J ( { ϕ ηh k ≥ l k } ∪ W ε ) − J ( { ϕ ηh k ≥ l k } ) + 1 h k Z W ε d { u hk ( · ,t k − h k ) ≥ l k − ε } ( x ) dx ≤ . By (7.19), { u h k ( · , t k − h k ) ≥ l k − ε } ⊆ { ϕ ( · , t k − h k ) ≥ l k − c k − ε } , so that we alsohave(7.22) J ( { ϕ ηh k ≥ l k } ∪ W ε ) − J ( { ϕ ηh k ≥ l k } )+ 1 h k Z W ε d { ϕ ( · ,t k − h k ) ≥ l k − c k − ε } ( x ) dx ≤ . If instead { u h k ( · , t k ) ≥ l k − ε } is unbounded, then inequality (7.21) must bereplaced by J ( { u h k ( · , t k ) ≥ l k − ε } ) + 1 h k Z { u hk ( · ,t k ) ≥ l k − ε }∩ B R d { u hk ( · ,t k − h k ) ≥ l k − ε } ( x ) dx ≤ J ( { u h k ( · , t k ) ≥ l k − ε } ∩ { ϕ ηh k ≥ l k } )+ 1 h k Z { u hk ( · ,t k ) ≥ l k − ε }∩{ ϕ ηhk ≥ l k }∩ B R d { u hk ( · ,t k − h k ) ≥ l k − ε } ( x ) dx , for R sufficiently large, see (7.10). Then, arguing as before, one obtains again (7.22).Notice that for z ∈ W ε we have(7.23) l k − ε < ϕ ( z, t k ) + c k + η Q ( z − x k ) < l k . Since, in turn, ϕ ( z, t k ) + c k ≥ l k − ε it follows that η Q ( z − x k ) < ε and thus, for ε small enough,(7.24) W ε ⊆ B C √ ε ( x k ) . Moreover, for every z ∈ W ε (7.25) ϕ ( z, t k − h k ) = ϕ ( z, t k ) − h k ∂ t ϕ ( z, t k ) + h k Z (1 − s ) ∂ tt ϕ ( z, t k − sh k ) ds . Let y be a point of minimal distance from z such that ϕ ( y, t k − h k ) = l k − c k − ε .Then, | z − y | = | d { ϕ ( · ,t k − h k ) ≥ l k − c k − ε } ( z ) | , and(7.26) ( z − y ) · Dϕ ( y, t k − h k ) = ±| z − y || Dϕ ( y, t k − h k ) | , ONLOCAL CURVATURE FLOWS 53 with a ‘+’ if ϕ ( z, t k − h k ) > l k − c k − ε and a ‘ − ’ else, so that the sign is oppositeto the sign of d { ϕ ( · ,t k − h k ) ≥ l k − c k − ε } ( z ). Hence,(7.27) ϕ ( z, t k − h k ) = ϕ ( y, t k − h k ) + ( z − y ) · Dϕ ( y, t k − h k )+ Z (1 − s )( D ϕ ( y + s ( z − y ) , t k − h k )( z − y )) · ( z − y ) ds = l k − c k − ε − d { ϕ ( · ,t k − h k ) ≥ l k − c k − ε } ( z ) | Dϕ ( y, t k − h k ) | + Z (1 − s )( D ϕ ( y + s ( z − y ) , t k − h k )( z − y )) · ( z − y ) ds . By (7.23) we deduce in particular ϕ ( x, t k ) + c k < l k , i.e.,(7.28) − ϕ ( x, t k ) ≥ c k − l k . Combining (7.28), (7.25), and (7.27), we deduce d { ϕ ( · ,t k − h k ) ≥ l k − c k − ε } ( z ) | Dϕ ( y, t k − h k ) |≥ − ε + h k ∂ t ϕ ( z, t k ) − h k Z (1 − s ) ∂ tt ϕ ( z, t k − sh k ) ds + Z (1 − s )( D ϕ ( y + s ( z − y ) , t k − h k )( z − y )) · ( z − y ) ds . Note that, in view of (7.23), | ϕ ( z, t k ) − ϕ ( y, t k ) | ≤ ε + Ch k = O ( h k ), provided that ε << h k are small enough. In turn, by (7.26) as | Dϕ ( y, t k − h k ) | is bounded awayfrom zero, we have | z − y | = O ( h k ) and, using also (7.24), we deduce(7.29) 1 h k d { ϕ ( · ,t k − h k ) ≥ l k − c k − ε } ( z ) ≥ ∂ t ϕ ( z, t k ) − εh k + O ( h k ) | Dϕ ( y, t k − h k ) | = ∂ t ϕ ( x k , t k ) + O ( √ ε ) − εh k + O ( h k ) | Dϕ ( x k , t k ) | + O ( √ ε ) + O ( h k ) . We now focus on the term J ( { ϕ ηh k ≥ l k } ∪ W ε ) − J ( { ϕ ηh k ≥ l k } )of inequality (7.22). Thanks to (4.5), if ε is small enough we know that(7.30) J ( { ϕ ηh k ≥ l k } ∪ W ε ) − J ( { ϕ ηh k ≥ l k } ) ≥ | W ε | ( κ ( x k , { ϕ ηh k ≥ ϕ ηh k ( x k ) } ) − o ε (1)) , recalling that ϕ ηh k ( x k ) is not a critical value of ϕ ηh k .Using therefore (7.22), (7.29) and (7.30), dividing by | W ε | and sending ε → η > ∂ t ϕ ( x k , t k ) + O ( h k ) | Dϕ ( x k , t k ) | + O ( h k ) + κ ( x k , { ϕ ηh k ≥ ϕ ηh k ( x k ) } ) ≤ . Letting simultaneously η → k → ∞ and using Lemma 2.8 we deduce (7.18). Step 2. Now we consider the case Dϕ (¯ z ) = 0 and we show that ϕ t (¯ z ) ≤ 0. Let ψ n be defined as in (2.16) and let z n = ( x n , t n ) be a sequence of maximizers of u − ψ n , such that x n → ¯ x and t n → ¯ t − . If x n = ¯ x for a (not relabeled) subsequence,then (for large n ) Dψ n ( x n , t n ) = 0 and (7.18) holds for ψ n at z n . Passing to the limit and using the properties of f (where f is the function appearing in (2.16)),we deduce that ∂ϕ∂t (¯ z ) ≤ z n = (¯ x, t n ) for all n sufficiently large. Set b n := ¯ t − t n and set(7.31) r n := f − ( a n b n ) , where a n → T ∗ ( r n ) of the solution of (7.17)with r replaced by r n , satisfies T ∗ ( r n ) ≥ b n for n large enough. To show thatsuch a choice for a n is possible, set g ( t ) = sup ≤ s ≤ t ˆ c ( f − ( s )) f ′ ( f − ( s )) , and notice that g ( t ) ≤ ˆ c ( t ) for t small, it is non decreasing in t , and g ( t ) → t → T ∗ ( r n ) b n ≥ b n Z r n r n / c ( r ) = 1 b n Z f − ( a n b n ) f − ( a n b n / c ( r )= a n − Z a n b n a n b n / c ( f − ( s )) f ′ ( f − ( s )) ds ≥ a n g ( b n ) = 2 . where the last equality holds if we choose a n = 4 g ( b n ) → ψ n , we have that B (¯ x, r n ) ⊂ { ψ n ( · , t n ) ≤ ψ n (¯ x, t n ) + 2 f ( r n ) }⊂ { u ( · , t n ) ≤ u (¯ x, t n ) + 2 f ( r n ) } . Note that the last inclusion follows from the maximality of u − ψ n at z n and thefact that u ( z n ) = ψ n ( z n ). By (7.32) and Proposition 7.15,¯ x ∈ { u ( · , ¯ t ) ≤ u (¯ x, t n ) + 2 f ( r n ) } . Thus, using also the maximality of u − ϕ at ¯ z , and recalling (7.31), we have ϕ (¯ x, t n ) − ϕ (¯ z ) − b n ≤ u (¯ x, t n ) − u (¯ x, ¯ t ) − b n ≤ f ( r n ) − b n = − a n . Passing to the limit, we conclude that ∂ t ϕ (¯ z ) ≤ (cid:3) Perimeter descent. In this part we address the problem of the perimeterdescent for variational curvature flows. The results refer to any viscosity solution u : R N × [0 , T ] → R to (2.4), where k is the first variation of a generalized perimeterin the sense of Definition 5.1. Throughout this subsection we also assume that theadditional conditions stated in Subsection 3.1 and 3.2 hold, so that such a solutionis unique and coincides with the one built through the minimizing movements.First, we generalize to our setting a fact that is well known in the context ofthe mean curvature flow: whenever there is no fattening, the perimeter decreasesin time. Proposition 7.18. Let ≤ t ≤ t ≤ T , let λ ∈ R and assume that |{ u ( · , t ) = λ }| = 0 . Then, J ( { u ( · , t ) > λ } ) ≤ J ( { u ( · , t ) > λ } ) . ONLOCAL CURVATURE FLOWS 55 Proof. Set ˜ u := d { u ( · ,t ) >λ } , and let ˜ u : [0 , T − t ] → R be the viscosity solution of(2.4) with initial condition ˜ u . By Remark 3.9, we get { ˜ u ( · , t ) > λ } = { u ( · , t + t ) > λ } for every t ∈ [0 , T − t ]. Let now ˜ u h be the approximate solution defined in (7.13).Then, by Proposition 7.12 we have J ( { ˜ u h ( · , t − t )) > λ } ≤ J ( { ˜ u h ( · , > λ } . Since ˜ u h → ˜ u pointwise (indeed, uniformly) and since |{ ˜ u ( · , t − t ) = λ }| = |{ u ( · , t ) = λ }| = 0 , we easily deduce that { ˜ u h ( · , t − t )) > λ } → { ˜ u ( · , t − t )) > λ } = { u ( · , t )) > λ } in measure, as h → ∞ . By the lower semicontinuity of J we conclude(7.33) J ( { u ( · , t )) > λ } ) ≤ lim inf h J ( { ˜ u h ( · , t − t )) > λ }≤ J ( { ˜ u h ( · , > λ } = J ( { u ( · , t )) > λ } . (cid:3) Remark 7.19. A natural question is whether the assumption of Proposition 7.18is satisfied. It seems reasonable to believe that, whenever the initial set { u > λ } is smooth enough, then there are no flat levels along the flow. To our knowledge,such a result is not known even for the canonical mean curvature flow. On theother hand, if the initial set E is star shaped, one can build u such that all itssuperlevels are homothetic to E . In view of the homogeneity properties of themean curvature and of the geometric evolution equation (2.4), all the superlevelsevolve staying homothetic to each other. As a consequence, superlevels are neverflat, and in turn the perimeter decreases along the flow. This is the case whenevera generalized curvature is homogeneous with respect to dilations, i.e. there exists α > κ ( x, lE ) = l − α κ ( x, E ) for every l > E ∈ C .Finally, we introduce a relaxed perimeter, defined on open sets, that alwaysdecreases along the flow. Definition 7.20. For every open set A ⊂ R N with compact boundary set ˜ J ( A ) :=inf lim inf J ( A n ) where the infimum is taken among all sequences of open sets A n with ¯ A n ∈ C , ¯ A n ⊂ A and R N \ A n → R N \ A in the Hausdorff sense. Remark 7.21. By the lower semicontinuity property of J , we have ˜ J ( A ) ≥ J ( A )for every open set A with compact boundary. The converse inequality is in generalfalse. For instance let J be the standard perimeter and let A := B \ { xy = 0 } .Then, J ( A ) = J ( B ), while it is easy to see that ˜ J ( A ) = J ( B ) + 4. It is well known(see [18]) that if u = d A , then the level-set { u ( · , t ) = 0 } is fat for every positivetime. Moreover, lim t → J ( { u ( · , t ) > } ) = ˜ J ( A ) . In particular, the perimeter J (instantaneously) increases along the geometric flow.The example somewhat motivates Definition 7.20. As we will see, the relaxedperimeter ˜ J instead is always non increasing. Remark 7.22. Clearly, in Definition 7.20 we can always assume that, whenever A is bounded, A n are compactly contained in A n +1 for every n (and a similarcondition for unbounded sets). Moreover, we can remove the regularity assumptionon A n without affecting the notion of ˜ J . Indeed, let ˆ J be defined as in Definition7.20, but without the requirement of the C ℓ,β -regularity. Clearly ˆ J ≤ ˜ J . To provethe converse inequality, consider an optimal sequence of open sets ˆ A n such that J ( ˆ A n ) → ˆ J ( A ). It is enough to regularize each A n in order to have an optimalsequence ˜ A n for ˜ J . This can be easily done in view of Lemma 4.1. The details areleft to the reader.We state a lemma which clarifies the role of Definition 7.20 in the viscosityapproach to geometric flows. For the reader convenience, we omit its technical butstraightforward proof, Lemma 7.23. Let A ⊂ R N be open with compact boundary, and let ¯ A n ∈ C with ¯ A n ⊂ A n compactly contained in for every n .Then, there exists a one-Lipschitz function u A and a sequence λ n → such that A = { u A > } ; A n = { u A > λ n } ; u A = d A n + c n in a neighborhood of ∂A n , for some suitable constant c n . Proposition 7.24. The relaxed perimeter ˜ J decreases along the geometric flow.More precisely, for every λ ∈ R the function t → ˜ J ( { u ( · , t ) > λ } ) is not increasing.Proof. To easy notations, we will assume λ = 0. Let 0 ≤ t ≤ t ≤ T . We have toprove that ˜ J ( { u ( · , t ) > } ) ≤ ˜ J ( { u ( · , t ) > } ) . Let ( A n ) be an optimal sequence for Definition 7.20 with A replaced by A := { u ( · , t ) > } . Clearly, we may assume that ¯ A n ⊂ A n +1 for every n . Moreover,let λ n , u A be as in Lemma 7.23. By property 3) of Lemma 7.23, we have that thefunction that associate to λ the corresponding superlevel set of u A is continuousfrom a neighborhood of each λ n to C . In particular, the function λ → J ( { u A ( · ) >λ ) } ) is continuous at each λ n . Notice that all except countably many levels of u A have null measure. Therefore, there exists a sequence ˜ λ n → |{ u A ( · ) = ˜ λ n ) }| = 0 for every n ;ii) J ( { u A ( · ) > ˜ λ n } ) → ˜ J ( { u ( · , t ) > λ } ) as n → ∞ .Let ˜ u : [0 , T − t ] → R be the solution to (2.4) with initial condition u A . ByProposition 7.18 we have J ( { ˜ u ( · , t − t ) > ˜ λ n } ) ≤ J ( { ˜ u ( · , > ˜ λ n } ) . 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Springer-Verlag, Berlin, 1992. Shape sen-sitivity analysis.[30] Enrico Valdinoci. A fractional framework for perimeters and phase transitions. Milan J. Math. ,81(1):1–23, 2013.(Antonin Chambolle) CMAP, Ecole Polytechnique, CNRS, France E-mail address , A. Chambolle: [email protected] (Massimiliano Morini) Dip. di Matematica, Univ. Parma, Italy E-mail address , M. Morini: [email protected] (Marcello Ponsiglione) Dip. di Matematica, Univ. Roma-I “La Sapienza”, Roma, Italy E-mail address , M. Ponsiglione:, M. Ponsiglione: