Local behavior of fractional p -minimizers
aa r X i v : . [ m a t h . A P ] M a y LOCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS AGNESE DI CASTRO, TUOMO KUUSI, AND GIAMPIERO PALATUCCI
Abstract.
We extend the De Giorgi-Nash-Moser theory to nonlocal, pos-sibly degenerate integro-differential operators.
To appear in
Ann. Inst. H. Poincare Anal. Non Lineaire . Introduction
The aim of this paper is to develop localization techniques in order to establishregularity results for nonlocal integro-differential operators and minimizers of frac-tional order s ∈ (0 ,
1) and summability p >
1. Let Ω be a bounded domain and let g be a function in the fractional Sobolev space W s,p ( R n ). We shall prove general localregularity estimates for the minimizers u , where u is minimizing the functional(1.1) F ( v ) := Z R n Z R n K ( x, y ) | v ( x ) − v ( y ) | p d x d y, over the class of functions { v ∈ W s,p ( R n ) : v = g a.e. in R n \ Ω } . Here K is asuitable symmetric kernel of order ( s, p ) with just measurable coefficients, see (2.1).It is standard to show, which is in fact our Theorem 2.3 below, that minimizerscan be equivalently characterized by the weak solutions to the following class ofintegro-differential problems(1.2) ( L u = 0 in Ω ,u = g in R n \ Ω , where the operator L is defined formally by(1.3) L u ( x ) = P. V. Z R n K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) d y, x ∈ R n ;the symbol P. V. means “in the principal value sense”. We immediately refer toSection 2 for the precise assumptions on the involved quantities.To simplify, one can keep in mind the model case when the kernel K ( x, y ) coincideswith | x − y | − ( n + sp ) , though in such a case the difficulties arising from having merely Mathematics Subject Classification.
Primary 35D30, 35B45; Secondary 35B05, 35R05,47G20, 60J75.
Key words and phrases.
Quasilinear nonlocal operators, fractional Sobolev spaces, H¨older regu-larity, Caccioppoli estimates, singular perturbations. measurable coefficients disappear; that is, the function u reduces to the solution ofthe following problem(1.4) ( ( − ∆) sp u = 0 in Ω ,u = g in R n \ Ω , where the symbol ( − ∆) sp denotes the standard fractional p -Laplacian operator.Recently, a great attention has been focused on the study of problems involvingfractional Sobolev spaces and corresponding nonlocal equations, both from a puremathematical point of view and for concrete applications, since they naturally arisein many different contexts. For an elementary introduction to this topic and for aquite extensive list of related references we refer to [8].However, for what concerns regularity and related results for this kind of operatorswhen p = 2, the theory seems to be rather incomplete. Nonetheless, some partialresults are known. Firstly, we would like to cite the higher regularity contributionsfor viscosity solutions in the case when s is close to p is suitably large - thus falling in the Morrey embeddingcase when concerning regularity. See also [9] for some basic results for fractional p -eigenvalues.On the contrary, when p = 2 and K ( x, y ) = | x − y | − n − s , that is the case of thewell-known fractional Laplacian operator ( − ∆) s , the situation simplifies notably.Although having been a classical topic in Functional and Harmonic Analysis as wellas in Partial Differential Equations for a long time, in the last years the growinginterest for such operator has become really significant and many important resultsfor the minimizer of (1.1) have been achieved. For what concerns the main topic inthe present paper, i.e., the local behavior of the fractional minimizers, it is worthmentioning the very relevant contributions for the case p = 2 by Kassmann ([12,13]); see also [31, 30]. In particular, among other results, Kassmann proves H¨olderregularity and a Harnack inequality “revisited” in the right form taking into accountthe nonlocality of the fractional Laplacian operator; we refer also to [11] to discoverhow the classic Harnack inequality fails in the fractional framework.In the present paper, we will deal with a larger class of operators with a symmetrickernel K having only measurable coefficients, and, above all, satisfying fractionaldifferentiability for any s ∈ (0 ,
1) and p -summability for any p >
1. For this,we will have to handle not only the usual nonlocal character of such fractionaloperators, but also the difficulties given by the corresponding nonlinear behavior.As a consequence, we can make use neither of the powerful framework providedby the Caffarelli-Silvestre s -harmonic extension ([4]) nor of various tools as, e. g.,the sharp 3-commutators estimates introduced in [5] to deduce the regularity ofweak fractional harmonic maps, the strong barriers and density estimates in [26,28, 29], the commutator and energy estimates in [25, 27], and so on. Indeed, theaforementioned tools seem not to be trivially adaptable to a nonlinear framework;also, increasing difficulties are due to the non-Hilbertian structure of the involvedfractional Sobolev spaces W s,p when p is different than 2. OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 3 We will have to work carefully in order to obtain the needed local estimates. Forthis, we want to underline that a specific quantity will be fundamental throughoutthe whole paper. Namely, we introduce the nonlocal tail of a function v ∈ W s,p ( R n )in the ball B R ( x ) ⊂ R n given by(1.5) Tail( v ; x , R ) := " R sp Z R n \ B R ( x ) | v ( x ) | p − | x − x | − ( n + sp ) d x p − . Note that the above number is finite by H¨older’s inequality whenever v ∈ L q ( R n ), q ≥ p −
1, and
R >
0. As expected, the way how the nonlocal tail will be managedis a key-point in the present extended local theory. We believe that this is a generalfact that will have to be taken into account in other results and extensions in thenonlinear fractional framework.We are now ready to introduce our main results. The first one describes the localboundedness.
Theorem 1.1 ( Local boundedness).
Let p ∈ (1 , ∞ ) , let u ∈ W s,p ( R n ) be a weaksubsolution to problem (1.2) and let B r ≡ B r ( x ) ⊂ Ω . Then the following estimateholds true sup B r/ ( x ) u ≤ δ Tail( u + ; x , r/
2) + cδ − ( p − nsp − Z B r ( x ) u p + d x ! p , (1.6) where Tail( u + ; x , r/ is defined in (1.5) , u + = max { u, } is the positive part ofthe function u , δ ∈ (0 , , and the constant c depends only on n, p, s and on thestructural constants λ, Λ defined in (2.1) . The parameter δ allows interpolation between the local and nonlocal terms. Armedwith the Logarithmic Lemma and the Caccioppoli estimate with tail introduced be-low, together with the deduced local boundedness, we can prove our main result,that is, the H¨older continuity theorem. Theorem 1.2 ( H¨older continuity).
Let p ∈ (1 , ∞ ) and let u ∈ W s,p ( R n ) be asolution to problem (1.2) . Then u is locally H¨older continuous in Ω . In particu-lar, there are positive constants α , α < sp/ ( p − , and c , both depending only on n, p, s, λ, Λ , such that if B r ( x ) ⊂ Ω , then osc B ̺ ( x ) u ≤ c (cid:16) ̺r (cid:17) α Tail( u ; x , r ) + − Z B r ( x ) | u | p d x ! p holds whenever ̺ ∈ (0 , r ] . The theorem above provides an extension of classical analogous results by DeGiorgi-Nash-Moser ([6, 24, 23]) to the nonlocal, nonlinear framework. It also extendsthe recent aforementioned result by Kassmann ([12]) to the case p = 2. Moreover,it is worth noticing that in the linear case studied in [12] a further boundednessassumption is required, which is now for free thanks to Theorem 1.1. A. DI CASTRO, T. KUUSI, AND G. PALATUCCI
In the proof of the H¨older continuity the following logarithmic estimate plays thekey role. We state it in the introduction as we think that it might be extremelyuseful also in other contexts.
Lemma 1.3 ( Logarithmic Lemma).
Let p ∈ (1 , ∞ ) . Let u ∈ W s,p ( R n ) be a weaksupersolution to problem (1.2) such that u ≥ in B R ≡ B R ( x ) ⊂ Ω . Then thefollowing estimate holds for any B r ≡ B r ( x ) ⊂ B R/ ( x ) and any d > , Z B r Z B r K ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) d + u ( x ) d + u ( y ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p d x d y ≤ c r n − sp n d − p (cid:16) rR (cid:17) sp [Tail( u − ; x , R )] p − + 1 o , (1.7) where Tail( u − ; x , R ) is defined in (1.5) , u − = max {− u, } is the negative part ofthe function u , and c depends only on n , p , s , λ and Λ . Then, we will show that the fractional p -minimizers, equivalently the weak so-lutions to the Euler-Lagrange equation associated to (1.1), satisfy the followingnonlocal Caccioppoli-type inequalities. Theorem 1.4 ( Caccioppoli estimates with tail).
Let p ∈ (1 , ∞ ) and let u ∈ W s,p ( R n ) be a weak solution to problem (1.2) . Then, for any B r ≡ B r ( x ) ⊂ Ω andany nonnegative φ ∈ C ∞ ( B r ) , the following estimate holds true Z B r Z B r K ( x, y ) | w ± ( x ) φ ( x ) − w ± ( y ) φ ( y ) | p d x d y ≤ c Z B r Z B r K ( x, y )(max { w ± ( x ) , w ± ( y ) } ) p | φ ( x ) − φ ( y ) | p d x d y (1.8) + c Z B r w ± ( x ) φ p ( x ) d x sup y ∈ supp φ Z R n \ B r K ( x, y ) w p − ± ( x ) d x ! , where w ± := ( u − k ) ± and c depends only on p .Remark . The estimate in (1.8) continues to hold for w + when u is merely a weaksubsolution to (2.3) and for w − when u is a weak supersolution to (2.3).Notice that, as expected, in the nonlocal framework one has to take into accounta suitable tail; see, in particular, the estimate in (5.14) below to see how the secondterm in the right hand-side of (1.8) is controlled by a tail as given in definition (1.5).Also, it is worth mentioning that other fractional Caccioppoli-type inequalities havebeen recently used in different contexts (see, for instance, [21, 22, 9]), although noneof them takes into account the tails. Let us finally comment some recent results in the literature. In [7] we proveHarnack-type inequalities with tail for weak supersolutions and solutions to (1.2).These can be applied to obtain H¨older continuity of the solutions. However, the We recently discovered that Kassmann proved similar Caccioppoli estimates with tail terms inthe linear case, when p = 2; see [10]. OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 5 proof in [7] are heavily based on the tools developed in the present paper. Moreover,the regularity theory for the inhomogeneous counterpart L u = f have been settledin [14] in a general setting, including also the case when the source term f is merelya measure. In turn, these results are partly based on the quantitative estimatesestablished here. The principal value definition (1.3) has been used in [17] to obtainregularity results in the context of viscosity solutions. Also, for general existenceresults and other regularity issues, we refer to the very recent contributions in [15],and in [2], where the related fractional p -eigenvalue problem has been considered.The paper is organized as follows. In Section 2 below, we fix the notation byalso providing some preliminary results. Section 3 is devoted to the proof of theLog Lemma 1.3 and the Caccioppoli estimates with tail in Theorem 1.4. In Section 4,we establish the local boundedness given by Theorem 1.1. In Section 5, we shallfinally prove the H¨older continuity given by Theorem 1.2.2. Preliminaries
In this section, we state the general assumptions of the problem we deal with inthe present paper, we fix notation, and we provide some definitions and some basicpreliminary results that we will use in the following pages.The kernel K : R n × R n → [0 , ∞ ) is a symmetric measurable function such that(2.1) λ ≤ K ( x, y ) | x − y | n + sp ≤ Λ for almost every x, y ∈ R n , for some s ∈ (0 , p >
1, Λ ≥ λ ≥
1. Notice that such assumption on K can beweakened as follows λ ≤ K ( x, y ) | x − y | n + sp ≤ Λ for almost every x, y ∈ R n s. t. | x − y | ≤ , ≤ K ( x, y ) | x − y | n + η ≤ M for almost every x, y ∈ R n s. t. | x − y | > , for some s, λ, Λ as above, η > M ≥
1; see, e. g., [12, 13]. For the sake ofsimplicity, we will keep the assumption in (2.1), since such a choice will imply norelevant differences in all the proofs in the rest of the paper.For any p ∈ [1 , ∞ ) and s ∈ (0 ,
1) we denote by W s,p ( R n ) the fractional Sobolevspace, that is W s,p ( R n ) := ( v ∈ L p ( R n ) : | v ( x ) − v ( y ) || x − y | np + s ∈ L p ( R n × R n ) ) ;i. e., an intermediary Banach space between L p ( R n ) and W ,p ( R n ), endowed withthe natural norm k v k W s,p ( R n ) := (cid:18)Z R n | v | p d x (cid:19) p + (cid:18)Z R n Z R n | v ( x ) − v ( y ) | p | x − y | n + sp d x d y (cid:19) p . In a similar way, it is possible to define the fractional Sobolev space W s,p (Ω) in adomain Ω ⊆ R n . Furthermore, by saying that v belongs to W s,p (Ω) we mean that v ∈ W s,p ( R n ) and v = 0 almost everywhere in R n \ Ω. A. DI CASTRO, T. KUUSI, AND G. PALATUCCI
As mentioned in the introduction, we define the nonlocal tail of a function v in theball B R ( x ), a quantity which will play an important role in the rest of the paper.For any v ∈ W s,p ( R n ) and B R ( x ) ⊂ R n , we write(2.2) Tail( v ; x , R ) := " R sp Z R n \ B R ( x ) | v ( x ) | p − | x − x | − n − sp d x p − , which is a finite number by H¨older’s inequality since v ∈ L p ( R n ) and R > R n and g ∈ W s,p ( R n ), we will be interested inweak solutions to the following integro-differential problems(2.3) ( L u = 0 in Ω ,u = g in R n \ Ω , where the operator L is formally defined in (1.3). Notice that the boundary conditionis given in the whole complement of Ω, as usual when dealing with such nonlocaloperators. A model example we have in mind is the fractional p -Laplacian, that is − ( − ∆) sp u ( x ) = c ( n, s, p ) P. V. Z R n | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) | x − y | n + sp d x d y, with s ∈ (0 ,
1) and p > W s,p ( R n ), the following functional(2.4) F ( u ) = Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p d x d y. In view of the assumptions (2.1) on K , one can use the standard Direct Method toprove that there exists a unique p -minimizer of F over all u ∈ W s,p ( R n ) such that u ( x ) = g ( x ) for x ∈ R n \ Ω. Moreover, a p -minimizer u is a weak solution solutionto problem (2.3) and vice versa (see Theorem 2.3 below).To specify relevant spaces, for given g ∈ W s,p ( R n ), we define the convex sets of W s,p ( R n ) as K ± g (Ω) := { v ∈ W s,p ( R n ) : ( g − v ) ± ∈ W s,p (Ω) } and K g (Ω) := K + g (Ω) ∩ K − g (Ω) = { v ∈ W s,p ( R n ) : v − g ∈ W s,p (Ω) } . We recall that the functions in the space W s,p (Ω) are defined in the whole space,since they are considered to be extended to zero outside Ω.We conclude this section by recalling the definition of weak sub- and supersolu-tions as well as weak solutions to problem (2.3). Definition 2.1.
Let g ∈ W s,p ( R n ). A function u ∈ K − g ( K + g ) is a weak subsolution ( supersolution ) to problem (2.3) if(2.5) Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( η ( x ) − η ( y )) d x d y ≤ ( ≥ ) 0for every nonnegative η ∈ W s,p (Ω). OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 7 A function u is a weak solution to problem (2.3) if it is both weak sub- andsupersolution. In particular, u belongs to K g (Ω) and satisfies(2.6) Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( η ( x ) − η ( y )) d x d y = 0for every η ∈ W s,p (Ω).Similarly, we recall the definition of sub- and superminimizers of (2.4). We have Definition 2.2.
Let g ∈ W s,p ( R n ). A function u ∈ K − g is a subminimizer of thefunctional (2.4) over K − g if F ( u ) ≤ F ( u + η ) for every nonpositive η ∈ W s,p (Ω).Similarly, a function u ∈ K + g is a superminimizer of the functional (2.4) over K + g if F ( u ) ≤ F ( u + η ) for every nonnegative η ∈ W s,p (Ω).Finally, u ∈ K g is a minimizer of the functional (2.4) over K g if F ( u ) ≤ F ( u + η )for every η ∈ W s,p (Ω).2.1. Notation.
Before starting with the proofs, it is convenient to fix some nota-tion which will be used throughout the rest of the paper. Firstly, notice that we willfollow the usual convention of denoting by c a general positive constant which willnot necessarily be the same at different occurrences and which can also change fromline to line. For the sake of readability, dependencies of the constants will be oftenomitted within the chains of estimates, therefore stated after the estimate. Rele-vant dependences on parameters will be emphasized by using parentheses; specialconstants will be denoted by c , c ,....As customary, we denote by B R ( x ) = B ( x ; R ) := n x ∈ R n : | x − x | < R o the open ball centered in x ∈ R n with radius R >
0. When not important andclear from the context, we shall use the shorter notation B R := B ( x ; R ). We denoteby βB R the concentric ball scaled by a factor β >
0, that is βB R := B ( x ; βR ).Moreover, if f ∈ L ( S ) and the n -dimensional Lebesgue measure | S | of the set S ⊆ R n is finite and strictly positive, we write(2.7) ( f ) S := − Z S f ( x ) d x = 1 | S | Z S f ( x ) d x. Let k ∈ R , we denote by(2.8) w + ( x ) := ( u ( x ) − k ) + = max { u ( x ) − k, } , and(2.9) w − ( x ) := ( u ( x ) − k ) − = ( k − u ( x )) + . Clearly w + ( x ) = 0 in the set (cid:8) x ∈ S : u ( x ) > k (cid:9) , and w − ( x ) = 0 in the set (cid:8) x ∈ S : u ( x ) < k (cid:9) . A. DI CASTRO, T. KUUSI, AND G. PALATUCCI
Existence and uniqueness of the minimizers.
The proof of the existenceand uniqueness for fractional minimizers is simple and it is recorded into the follow-ing.
Theorem 2.3.
Let s ∈ (0 , and p ∈ [1 , ∞ ) , and let g ∈ W s,p ( R n ) . Then thereexists a minimizer u of (2.4) over K g . Moreover, if p > , then the solution isunique. Moreover, a function u ∈ K g is a minimizer of (2.4) over K g if and only ifit is a weak solution to problem (2.3) .Proof. The proof plainly follows by the Direct Method of Calculus of Variations.One can take any minimizing sequence u j ∈ K g . Due to the assumptions on thekernel K , one can control the fractional seminorm of u j , so that, one can find by pre-compactness in L p (see, for instance, [8, Theorem 6.7]) a subsequence u j k convergingpointwise a. e. to a function u ∈ K g . By Fatou’s Lemma we deduce that u is actuallya minimizer of (2.4) over K g . The uniqueness in the case p > u solves the corresponding Euler-Lagrange equationfollows by perturbing u ∈ K g with a test function in a standard way. Indeed,supposing that u ∈ K g is a minimizer of (2.4) over K g , take any φ ∈ W s,p (Ω) andcalculate formally ddt F ( u + tφ ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z R n Z R n K ( x, y ) dd t | u ( x ) − u ( y ) + t ( φ ( x ) − φ ( y )) | p d x d y (cid:12)(cid:12)(cid:12)(cid:12) t =0 = p Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( φ ( x ) − φ ( y )) d x d y . Since u is a minimizer, the term on the left is zero and hence u ∈ K g is a weaksolution to problem (2.3). For the converse, let u ∈ K g be a weak solution toproblem (2.3) and take φ = u − v ∈ W s,p (Ω), where v ∈ K g . Then, by Young’sinequality,0 = Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( φ ( x ) − φ ( y )) d x d y = Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p d x d y − Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( v ( x ) − v ( y )) d x d y ≥ p Z R n Z R n K ( x, y ) | u ( x ) − u ( y ) | p d x d y − p Z R n Z R n K ( x, y ) | v ( x ) − v ( y ) | p d x d y , and hence u is a minimizer of (2.4) over K g . (cid:3) OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 9 Fundamental estimates
In this section, we establish some relevant estimates that we will use in the fol-lowing. We believe that these results could have their own interest in the analysisof equations involving the (nonlinear) fractional Laplacian and related nonlocal op-erators. The first of them states a natural extension of the well-known Caccioppoliinequality to the nonlocal framework, by showing that in such a case one can takeinto account a suitable tail, in order to detect deeper informations.
Proof of Theorem 1.4.
For the sake of generality, we would point out that the presentproof is also valid when p = 1.Let u be a weak solution as in the statement. Testing (2.5) with η := w + φ p ,where φ is any nonnegative function in C ∞ ( B r ( x )), we get0 ≥ Z B r Z B r K ( x, y ) | u ( x ) − u ( y ) | p − (3.1) × ( u ( x ) − u ( y ))( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y )) d x d y +2 Z R n \ B r Z B r K ( x, y ) | u ( x ) − u ( y ) | p − × ( u ( x ) − u ( y )) w + ( x ) φ p ( x ) d x d y Note that η is an admissible test function since truncations of functions in W s,p ( R n )still belong to W s,p ( R n ).Let us consider the integrands of the two terms above separately. In the firstterm, we may assume without loss of generality that u ( x ) ≥ u ( y ); otherwise justexchange the roles of x and y below. We have | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y ))( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y ))= ( u ( x ) − u ( y )) p − (( u ( x ) − k ) + φ p ( x ) − ( u ( y ) − k ) + φ p ( y ))= ( w + ( x ) − w + ( y )) p − ( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y )) , u ( x ) , u ( y ) > k ( u ( x ) − u ( y )) p − w + ( x ) φ p ( x ) , u ( x ) > k , u ( y ) ≤ k , otherwise ≥ ( w + ( x ) − w + ( y )) p − ( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y )) . For the second term in the right hand-side of the inequality in (3.1) we instead have | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) w + ( x ) ≥ − ( u ( y ) − u ( x )) p − ( u ( x ) − k ) + ≥ − ( u ( y ) − k ) p − ( u ( x ) − k ) + = − w + ( y ) p − w + ( x ) , and estimating further we obtain Z R n \ B r Z B r K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) w + ( x ) φ p ( x ) d x d y ≥ − Z R n \ B r Z B r K ( x, y ) w p − ( y ) w + ( x ) φ p ( x ) d x d y ≥ − sup x ∈ supp φ Z R n \ B r K ( x, y ) w p − ( y ) d y ! Z B r w + ( x ) φ p ( x ) d x. We thus deduce from (3.1) that0 ≥ Z B r Z B r K ( x, y ) | w + ( x ) − w + ( y ) | p − (3.2) × ( w + ( x ) − w + ( y ))( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y )) d x d y − sup x ∈ supp φ Z R n \ B r K ( x, y ) w p − ( y ) d y ! Z B r w + ( x ) φ p ( x ) d x. Let us then consider the first term in the inequality above. If w + ( x ) ≥ w + ( y ) and φ ( x ) ≤ φ ( y ) in the integrand, we appeal to Lemma 3.1 below and get(3.3) φ p ( x ) ≥ (1 − c p ε ) φ p ( y ) − (1 + c p ε ) ε − p | φ ( x ) − φ ( y ) | p for any ε ∈ (0 ,
1] with the constant c p ≡ ( p − { , p − } ). Thus, by choosing ε := 1max { , c p } w + ( x ) − w + ( y ) w + ( x ) ∈ (0 , w + ( x ) − w + ( y )) p − w + ( x ) φ p ( x ) ≥ ( w + ( x ) − w + ( y )) p − w + ( x )(max { φ ( x ) , φ ( y ) } ) p −
12 ( w + ( x ) − w + ( y )) p (max { φ ( x ) , φ ( y ) } ) p − c (max { w + ( x ) , w + ( y ) } ) p | φ ( x ) − φ ( y ) | p with c ≡ c ( p ). Recall that in the estimate above we assumed that φ ( x ) ≤ φ ( y ),max { φ ( x ) , φ ( y ) } = φ ( y ). However, when 0 = w + ( x ) ≥ w + ( y ) ≥ w + ( x ) ≥ w + ( y )and φ ( x ) ≥ φ ( y ), the estimate in the display above is trivial and hence we concludethat it holds also in these cases. It follows that( w + ( x ) − w + ( y )) p − ( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y )) ≥ ( w + ( x ) − w + ( y )) p − ( w + ( x )(max { φ ( x ) , φ ( y ) } ) p − w + ( y ) φ p ( y )) −
12 ( w + ( x ) − w + ( y )) p (max { φ ( x ) , φ ( y ) } ) p − c (max { w + ( x ) , w + ( y ) } ) p | φ ( x ) − φ ( y ) | p ≥
12 ( w + ( x ) − w + ( y )) p (max { φ ( x ) , φ ( y ) } ) p − c (max { w + ( x ) , w + ( y ) } ) p | φ ( x ) − φ ( y ) | p whenever w + ( x ) ≥ w + ( y ). If, on the other hand, w + ( y ) > w + ( x ) in the integrand,we may interchange the roles of x and y in the display above by analogous reasoning.Hence we arrive in all cases at Z B r Z B r K ( x, y ) | w + ( x ) − w + ( y ) | p − × ( w + ( x ) − w + ( y ))( w + ( x ) φ p ( x ) − w + ( y ) φ p ( y )) d x d y OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 11 ≥ Z B r Z B r K ( x, y ) | w + ( x ) − w + ( y ) | p (max { φ ( x ) , φ ( y ) } ) p d x d y (3.4) − c Z B r Z B r K ( x, y )(max { w + ( x ) , w + ( y ) } ) p | φ ( x ) − φ ( y ) | p d x d y. Observing finally that | w + ( x ) φ ( x ) − w + ( y ) φ ( y ) | p ≤ p − | w + ( x ) − w + ( y ) | p (max { φ ( x ) , φ ( y ) } ) p +2 p − (max { w + ( x ) , w + ( y ) } ) p | φ ( x ) − φ ( y ) | p and combining this with (3.2) and (3.4) concludes the proof of (1.8) for w + .In order to prove the estimate in (1.8) for w − , it will suffice to proceed as above,using the function η = − w − φ , instead of η = w + φ , as a test function in the weakformulation of problem (2.3). (cid:3) Above we made use of the following trivial but very useful small lemma.
Lemma 3.1.
Let p ≥ and ε ∈ (0 , . Then | a | p ≤ | b | p + c p ε | b | p + (1 + c p ε ) ε − p | a − b | p , c p := ( p − { , p − } ) , holds for every a, b ∈ R m , m ≥ . Here Γ stands for the standard Gamma function.Proof. By the triangle inequality and convexity we obtain | a | p ≤ ( | b | + | a − b | ) p = (1 + ε ) p (cid:18)
11 + ε | b | + ε ε | a − b | ε (cid:19) p ≤ (1 + ε ) p − | b | p + (cid:18) εε (cid:19) p − | a − b | p . Estimating(1 + ε ) p − = 1 + ( p − Z ε t p − dt ≤ ε ( p −
1) max { , (1 + ε ) p − } , and then iterating, to get the Gamma function bound, concludes the proof. (cid:3) We would like to recall that, as in the classic local case, the proven
Caccioppoliestimates with tail encode basically all the informations deriving from the minimumproperty of the functions u for what concerns the corresponding H¨older continuity.We next show the validity of the second main tool, that is the LogarithmicLemma
Proof of Log Lemma 1.3.
Let d > φ ∈ C ∞ ( B r/ ) besuch that 0 ≤ φ ≤ , φ ≡ B r and | Dφ | < c r − in B r ⊂ B R/ . We use in the weak formulation of problem (2.3), the test function η defined by η = ( u + d ) − p φ p . Note that since u ≥ φ , the test function is well-defined. We get0 = Z B r Z B r K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) × (cid:20) φ p ( x )( u ( x ) + d ) p − − φ p ( y )( u ( y ) + d ) p − (cid:21) d x d y +2 Z R n \ B r Z B r K ( x, y ) | u ( x ) − u ( y ) | p − u ( x ) − u ( y )( u ( x ) + d ) p − φ p ( x ) d x d y =: I + I . (3.5)If u ( x ) > u ( y ), for the integrand of I , we use the inequality in Lemma 3.1, bychoosing there a = φ ( x ), b = φ ( y ) and ε = δ u ( x ) − u ( y ) u ( x ) + d ∈ (0 , , with δ ∈ (0 , , since u ( y ) ≥ y ∈ B r ⊂ B R . It follows that K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) (cid:20) φ p ( x )( u ( x ) + d ) p − − φ p ( y )( u ( y ) + d ) p − (cid:21) ≤ K ( x, y ) ( u ( x ) − u ( y )) p − ( u ( x ) + d ) p − φ p ( y ) " c δ u ( x ) − u ( y ) u ( x ) + d − (cid:18) u ( x ) + du ( y ) + d (cid:19) p − + c δ − p K ( x, y ) | φ ( x ) − φ ( y ) | p , where c ≡ c ( p ). Observe that the first term that appears in the right-hand side ofthe previous inequality can be rewritten as(3.6) K ( x, y ) (cid:18) u ( x ) − u ( y ) u ( x ) + d (cid:19) p φ p ( y ) − (cid:16) u ( y )+ du ( x )+ d (cid:17) − p − u ( y )+ du ( x )+ d + c δ =: J . Now, consider the real function t g ( t ) given by g ( t ) := 1 − t − p − t = − p − − t Z t τ − p d τ, ∀ t ∈ (0 , . We have that g is an increasing function in t , since t − t Z t τ − p d τ is a decreasing function (recall that p > g ( t ) ≤ − ( p − ∀ t ∈ (0 , . Moreover, if t ≤ /
2, then g ( t ) ≤ − p − p t − p − t . Therefore, if t = u ( y ) + du ( x ) + d ∈ (0 , / OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 13 that is, u ( y ) + d ≤ u ( x ) + d , then, since ( u ( x ) − u ( y ))( u ( y ) + d ) p − / ( u ( x ) + d ) p ≤
1, we get(3.7) J ≤ K ( x, y ) (cid:18) c δ − p − p (cid:19) (cid:20) u ( x ) − u ( y ) u ( y ) + d (cid:21) p − φ p ( y ) , Choosing(3.8) δ = p − p +1 c , we get J ≤ − K ( x, y ) p − p +1 (cid:20) u ( x ) − u ( y ) u ( y ) + d (cid:21) p − If, on the other hand, t = u ( y ) + du ( x ) + d ∈ (1 / , , that is, u ( y ) + d > u ( x ) + d , then J ≤ K ( x, y ) [ cδ − ( p − (cid:20) u ( x ) − u ( y ) u ( x ) + d (cid:21) p φ p ( y ) , and so, by the choice of δ in (3.8), we finally get(3.9) J ≤ − K ( x, y ) (2 p +1 − p − p +1 (cid:20) u ( x ) − u ( y ) u ( x ) + d (cid:21) p φ p ( y ) . Furthermore, if 2( u ( y ) + d ) < u ( x ) + d , then(3.10) (cid:20) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:21) p ≤ c (cid:20) u ( x ) − u ( y ) u ( y ) + d (cid:21) p − holds with c ≡ c ( p ). On the other hand, if 2( u ( y ) + d ) ≥ u ( x ) + d , recalling that wehave assumed u ( x ) > u ( y ), then(3.11) (cid:20) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:21) p = (cid:20) log (cid:18) u ( x ) − u ( y ) u ( y ) + d (cid:19)(cid:21) p ≤ p (cid:18) u ( x ) − u ( y ) u ( x ) + d (cid:19) p , where we have usedlog(1 + ξ ) ≤ ξ, ∀ ξ ≥ , with ξ = u ( x ) − u ( y ) u ( y ) + d ≤ u ( x ) − u ( y )] u ( x ) + d . Thus, combining (3.6) with (3.7), (3.9), (3.10) and (3.11), we conclude with K ( x, y ) | u ( x ) − u ( y ) | p − ( u ( x ) − u ( y )) (cid:20) φ p ( x )( u ( x ) + d ) p − − φ p ( y )( u ( y ) + d ) p − (cid:21) ≤ − c K ( x, y ) (cid:20) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:21) p φ p ( y ) + c δ − p K ( x, y ) | φ ( x ) − φ ( y ) | p . Observe that when u ( x ) = u ( y ), then the estimate above holds trivially. If, onthe other hand, u ( y ) > u ( x ) we can again exchange the roles of x and y in thecomputations above. We finally get for the first term in (3.5) that I ≤ − c Z B r Z B r K ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p φ p ( y ) d x d y (3.12) + c Z B r Z B r K ( x, y ) | φ ( x ) − φ ( y ) | p d x d y for a constant c ≡ c ( p ) by the choice of δ .For the second contribution in (3.5), namely I , we can proceed as follows. Firstof all, notice that when y ∈ B R , u ( y ) ≥ u ( x ) − u ( y )) p − ( d + u ( x )) p − ≤ x ∈ B r , y ∈ B R . Moreover, when y ∈ R n \ B R ,( u ( x ) − u ( y )) p − ≤ p − [ u p − ( x ) + ( u ( y )) p − − ] for all x ∈ B r . Therefore, I ≤ Z B R \ B r Z B r K ( x, y )( u ( x ) − u ( y )) p − ( d + u ( x )) − p φ p ( x ) d x d y +2 Z R n \ B R Z B r K ( x, y )( u ( x ) − u ( y )) p − ( d + u ( x )) − p φ p ( x ) d x d y ≤ c Z R n \ B r Z B r K ( x, y ) φ p ( x ) d x d y (3.13) + cd − p Z R n \ B R Z B r K ( x, y )( u ( y )) p − − d x d y follows for c ≡ c ( p ). By the assumptions on K and the fact that the support of φ belongs to B r/ , we have(3.14) Z R n \ B r Z B r K ( x, y ) φ p ( x ) d x d y ≤ c sup x ∈ B r/ r n Z R n \ B r K ( x, y ) d y ≤ cr n − sp and Z R n \ B R Z B r K ( x, y )( u ( y )) p − − d x d y ≤ c | B r | Z R n \ B R ( u ( y )) p − − | y − x | n + sp d y ≤ c r n R sp [Tail( u − ; x , R )] p − , (3.15)where we also used that, for any x ∈ B r , y ∈ R n \ B R and 2 r ≤ R , | y − x || y − x | ≤ | x − x || x − y | ≤ rR − r ≤ . OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 15 By combining (3.13) with (3.14) and (3.15), we obtain I ≤ c Z B r Z B r K ( x, y ) | φ ( x ) − φ ( y ) | p d x d y + cr n − sp + c d − p r n R − sp [Tail( u − ; x , R )] p − , which, together with (3.12) in (3.5), yields Z B r Z B r K ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) u ( x ) + du ( y ) + d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p φ p ( y ) d x d y ≤ c Z B r Z B r K ( x, y ) | φ ( x ) − φ ( y ) | p d x d y (3.16) + c d − p r n R − sp [Tail( u − ; x , R )] p − + cr n − sp . Finally, in order to conclude the proof, we need the following estimate Z B r Z B r K ( x, y ) | φ ( x ) − φ ( y ) | p d x d y ≤ cr − p Z B r Z B r | x − y | − n + p (1 − s ) d x d y ≤ cp (1 − s ) r − sp | B r | , (3.17)where we used the bound from above on the kernel K and the fact that we areassuming | Dφ | ≤ c r − . The proof of (1.7) is finished. (cid:3) A first consequence of the Logarithmic Lemma is the following
Corollary 3.2.
Let p ∈ (1 , ∞ ) and let u ∈ W s,p ( R n ) be the solution to problem (2.3) such that u ≥ in B R ≡ B R ( x ) ⊂ Ω . Let a, d > , b > and define v := min (cid:8) (log( a + d ) − log( u + d )) + , log( b ) (cid:9) . Then the following estimate holds true, for any B r ≡ B r ( x ) ⊂ B R/ ( x ) , − Z B r | v − ( v ) B r | p d x ≤ c n d − p (cid:16) rR (cid:17) sp [Tail( u − ; x , R )] p − + 1 o , where Tail( u − ; x , R ) is defined by (2.2) and c depends only on n , p , s , λ and Λ .Proof. By the fractional Poincar´e type inequality (see, e. g., Proposition 5.1, For-mula (6.3) in [20]) and the assumption in (2.1) for K we get − Z B r | v − ( v ) B r | p d x ≤ c r sp − n Z B r Z B r K ( x, y ) | v ( x ) − v ( y ) | p d x d y with a constant c ≡ c ( n, p, s, λ, Λ). Now observe that v is a truncation of the sumof a constant and log( u + d ) and hence it follows that Z B r Z B r K ( x, y ) | v ( x ) − v ( y ) | p d x d y ≤ Z B r Z B r K ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) u ( y ) + du ( x ) + d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p d x d y. At this stage, in order to conclude, it just suffices to apply the estimate in (1.7). (cid:3) Local boundedness
In this section, we prove the local boundedness for the fractional p -minimizers ofthe functional (2.4), as stated in Theorem 1.1. Proof of Theorem 1.1.
Before starting, let us give some definitions. For any j ∈ N and r > B r ( x ) ⊂ Ω,(4.1) r j = 12 (1 + 2 − j ) r, ˜ r j = r j + r j +1 ,B j = B r j ( x ) , ˜ B j = B ˜ r j ( x ) . Moreover, take φ j ∈ C ∞ ( ˜ B j ) , ≤ φ j ≤ , φ j ≡ B j +1 , and | Dφ j | < j +3 /r,k j = k + (1 − − j )˜ k, ˜ k j = k j +1 + k j , ˜ k ∈ R + and k ∈ R . (4.2) ˜ w j = ( u − ˜ k j ) + and w j = ( u − k j ) + . By the fractional Poincar´e inequality applied to the function ˜ w j φ j , as definedabove, together with the properties of the kernel K , we plainly get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Z B j | ˜ w j ( x ) φ j ( x ) | p ∗ d x ! p ∗ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Z B j ˜ w j ( x ) φ j ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ c − Z B j | ˜ w j φ j − ( ˜ w j φ j ) B j | p ∗ d x ! pp ∗ (4.3) ≤ c r sp r n Z B j Z B j K ( x, y ) | ˜ w j ( x ) φ j ( x ) − ˜ w j ( y ) φ j ( y ) | p d x d y, where p ∗ = np/ ( n − sp ) is the critical exponent for fractional Sobolev embeddings,so that we are now dealing with the case when sp < n .Using the nonlocal Caccioppoli inequality with tail given by (1.8), with w + = ˜ w j and φ = φ j there, we arrive at − Z B j | ˜ w j ( x ) φ j ( x ) | p ∗ d x ! pp ∗ ≤ cr sp − Z B j Z B j K ( x, y )(max { ˜ w j ( x ) , ˜ w j ( y ) } ) p | φ j ( x ) − φ j ( y ) | p d x d y (4.4) + cr sp − Z B j ˜ w j ( y ) φ pj ( y ) d y sup y ∈ supp φ j Z R n \ B j K ( x, y ) ˜ w p − j ( x ) d x ! + c − Z B j | ˜ w j ( x ) φ j ( x ) | p d x. OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 17 By the definition of φ j and the assumption (2.1), we obtain the following estimatefor the first term in the right hand-side of the inequality above, r sp − Z B j Z B j K ( x, y )(max { ˜ w j ( x ) , ˜ w j ( y ) } ) p | φ j ( x ) − φ j ( y ) | p d x d y ≤ c jp r sp r p − Z B j w pj ( y ) Z B j d x | x − y | n − p (1 − s ) ! d y ≤ c jp p (1 − s ) − Z B j w pj ( x ) d x. (4.5)For the second term on the right in (4.4), we get c r sp − Z B j ˜ w j ( y ) φ pj ( y ) d y sup y ∈ supp φ j Z R n \ B j K ( x, y ) ˜ w p − j ( x ) d x ! ≤ c j ( n + sp ) r sp − Z B j w pj ( y )(˜ k j − k j ) p − d y ! Z R n \ B j w p − j ( x ) | x − x | n + sp d x ! ≤ c j ( n + sp + p − ˜ k p − [Tail( w ; x , r/ p − − Z B j w pj ( y ) d y, (4.6)where we have just used the definitions in (4.1)–(4.2), the facts that ˜ w j ≤ w pj / (˜ k j − k j ) p − and that y ∈ ˜ B j = supp φ j and x ∈ R n \ B j yield | x − x || x − y | ≤ | x − y | + | x − x || x − y | ≤ r j r j − ˜ r j ≤ j +4 . The left hand-side of (4.4) can be estimated from below as follows − Z B j | ˜ w j ( x ) φ j ( x ) | p ∗ d x ! pp ∗ ≥ ( k j +1 − ˜ k j ) ( p ∗− p ) pp ∗ − Z B j +1 w pj +1 ( x ) d x ! pp ∗ = ˜ k j +2 ! ( p ∗− p ) pp ∗ − Z B j +1 w pj +1 ( x ) d x ! pp ∗ . (4.7)By combining (4.4) with (4.5), (4.6) and (4.7), we obtain ˜ k − p/p ∗ ( j +2) ( p ∗− p ) p ∗ ! p A p p ∗ j +1 ≤ c j ( n + sp + p − p (1 − s ) + [Tail( w ; x , r/ p − ˜ k p − + 1 ! A pj , where we have set A j := (cid:16) − R B j w pj ( x ) d x (cid:17) p . Now, by taking(4.8) ˜ k ≥ δ Tail( w ; x , r/ , δ ∈ (0 , , we get(4.9) (cid:18) A j +1 ˜ k (cid:19) pp ∗ ≤ δ − pp ¯ c pp ∗ j (cid:16) n + sp + p − p + spn (cid:17) A j ˜ k , where ¯ c = 2 p ∗− p ) p c p ∗ p (2 + ( p (1 − s )) − ) p ∗ p . Setting β := sp/ ( n − sp ) = p ∗ /p − > C := 2 ( n + sp + p − np ( n − sp ) + spn − sp >
1, theestimate in (4.9) becomes A j +1 ˜ k ≤ δ (1 − p ) p ∗ p ¯ c C j (cid:18) A j ˜ k (cid:19) β Thus, it suffices to prove that the following estimate on A does hold A ˜ k ≤ δ ( p − p ∗ βp ¯ c − β C − β and, by a well-known iteration argument, it will follow A j → j → ∞ . Since( p − p ∗ β p = p − p nn − ps n − spsp = ( p − nsp , we choose ˜ k = δ Tail( w ; x , r/
2) + δ − ( p − nsp HA , with H := ¯ c β C β , which is in accordance with (4.8).We deducesup B r/ u ≤ k + ˜ k = k + δ Tail(( u − k ) + ; x , r/
2) + δ − ( p − nsp H (cid:18) − Z B r ( u − k ) p + (cid:19) p , which finally gives the desired result by taking k = 0.The remaining case, that is when sp = n , can be treated exactly as above, justreplacing p ⋆ by a suitable power q in the left hand-side of (4.3) and consequentlyadjusting the exponents in the rest of the proof. (cid:3) Remark . Similarly, it is possible to prove that the weak solutions to problem (2.3)are locally bounded from below, satisfying an estimate analogous to the one in (1.6).The proof is exactly as before: one has just to work with ˜ w j = (˜ k j − u ) + and w j = ( k j − u ) + instead of the auxiliary functions defined in (4.2) and make use ofthe corresponding Caccioppoli estimate (1.8) for w − .5. H¨older continuity
This section is devoted to the proof of the H¨older continuity of solutions, namelyTheorem 1.2. As in the local framework, an iteration lemma is the keypoint of theproof. However, as before, we have to handle the nonlocality of the involved operatorand thus a certain care is required. In the proof below, all the estimates proven inprevious sections will appear.
OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 19 Before starting, let us fix some notation. For any j ∈ N , let 0 < r < R/
2, forsome R such that B R ( x ) ⊂ Ω, r j := σ j r , σ ∈ (0 , /
4] and B j := B r j ( x ) . Moreover, let us define12 ω ( r ) = 12 ω (cid:16) r (cid:17) := Tail( u ; x , r/
2) + c (cid:18) − Z B r | u | p d x (cid:19) p , with Tail( u ; x , r/
2) as in (2.2) and c as in (1.6), and ω ( r j ) := (cid:18) r j r (cid:19) α ω ( r ) , for some α < spp − . In order to prove the Theorem 1.2, it will suffice to prove the following
Lemma 5.1.
Under the notation introduced above, let u ∈ W s,p ( R n ) be the solutionto problem (2.3) . Then (5.1) osc B j u ≡ sup B j u − inf B j u ≤ ω ( r j ) , ∀ j = 0 , , , .... Proof.
We will proceed by induction. For this, note that by the definition of ω ( r )and Theorem 1.1 (with δ = 1 there), the estimate in (5.1) trivially holds for j = 0,since, in particular, both the functions ( u ) + and ( − u ) + are weak subsolutions.Now, we make a strong induction assumption and assume that (5.1) is valid forall i ∈ { , . . . , j } for some j ≥
0, and then we prove that it holds also for j + 1. Wehave that either(5.2) | B j +1 ∩ { u ≥ inf B j u + ω ( r j ) / }|| B j +1 | ≥ . or(5.3) | B j +1 ∩ { u ≤ inf B j u + ω ( r j ) / }|| B j +1 | ≥ u j := u − inf B j u , and if (5.3) holds, we set u j := ω ( r j ) − ( u − inf B j u ). In all cases we have that u j ≥ B j and(5.4) | B j +1 ∩ { u j ≥ ω ( r j ) / }|| B j +1 | ≥ u j is a weak solution satisfying(5.5) sup B i | u j | ≤ ω ( r i ) ∀ i ∈ { , . . . , j } . We now claim that under the induction assumption we have(5.6) [Tail( u j ; x , r j )] p − ≤ c σ − α ( p − [ ω ( r j )] p − , where the constant c depends only on n, p, s and the difference of sp/ ( p −
1) and α ,but, in particular, it is independent of σ . Indeed, we have[Tail( u j ; x , r j )] p − = r spj j X i =1 Z B i − \ B i | u j ( x ) | p − | x − x | − n − sp d x + r spj Z R n \ B | u j ( x ) | p − | x − x | − n − sp d x ≤ r spj j X i =1 [ sup B i − | u j | ] p − Z R n \ B i | x − x | − n − sp d x + r spj Z R n \ B | u j ( x ) | p − | x − x | − n − sp d x ≤ c j X i =1 (cid:18) r j r i (cid:19) sp [ ω ( r i − )] p − , where on the last line we used (5.5) and Z R n \ B | u j ( x ) | p − | x − x | − n − sp d x ≤ cr − sp sup B | u | p − + cr − sp [ ω ( r )] p − + c Z R n \ B | u ( x ) | p − | x − x | − n − sp d x ≤ cr − sp [ ω ( r )] p − . Estimating further as j X i =1 (cid:18) r j r i (cid:19) sp [ ω ( r i − )] p − = [ ω ( r )] p − (cid:18) r j r (cid:19) α ( p − j X i =1 (cid:18) r i − r i (cid:19) α ( p − (cid:18) r j r i (cid:19) sp − α ( p − = [ ω ( r j )] p − σ − α ( p − j − X i =0 σ i ( sp − α ( p − ≤ [ ω ( r j )] p − σ − α ( p − − σ sp − α ( p − ≤ sp − α ( p − log(4)( sp − α ( p − σ − α ( p − [ ω ( r j )] p − , where we have used the fact that σ ≤ / α < sp/ ( p − c depending only on n, p, s and the difference of sp/ ( p −
1) and α .Next, consider the function v defined as follows(5.7) v := min ((cid:20) log (cid:18) ω ( r j ) / du j + d (cid:19)(cid:21) + , k ) , k > . OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 21 Applying then Corollary 3.2, obviously with a ≡ ω ( r j ) / b ≡ exp( k ), we get − Z B j +1 | v − ( v ) B j +1 | p d x ≤ c (cid:26) d − p (cid:18) r j +1 r j (cid:19) sp [Tail( u j ; x , r j )] p − + 1 (cid:27) . Thus, as a consequence of the estimate in (5.6), we arrive at − Z B j +1 | v − ( v ) B j +1 | p d x ≤ c n d − p σ sp − α ( p − [ ω ( r j )] p − + 1 o . Therefore, choosing d = ε ω ( r j ) with ε := σ spp − − α , we get(5.8) − Z B j +1 | v − ( v ) B j +1 | d x ≤ c, where the constant c depends only on n, p, s, λ, Λ and the difference of sp/ ( p − α .To continue, denote in short ˜ B ≡ B j +1 , and follow the path paved in [19, Lemma2.107], together with (5.4) and the definition of v given in (5.7). We obtain k = 1 | ˜ B ∩ { u j ≥ ω ( r j ) / }| Z ˜ B ∩ { u j ≥ ω ( r j ) / } k d x = 1 | ˜ B ∩ { u j ≥ ω ( r j ) / }| Z ˜ B ∩ { v =0 } k d x ≤ | ˜ B | Z ˜ B ( k − v ) d x = 2[ k − ( v ) ˜ B ] . By integrating the preceding inequality over the set ˜ B ∩ { v = k } we obtain | ˜ B ∩ { v = k }|| ˜ B | k ≤ | ˜ B | Z ˜ B ∩ { v = k } [ k − ( v ) ˜ B ] d x ≤ | ˜ B | Z ˜ B | v − ( v ) ˜ B | d x ≤ c, thanks to (5.8). Let us take k = log (cid:18) ω ( r j ) / ε ω ( r j )3 ε ω ( r j ) (cid:19) = log (cid:18) / ε ε (cid:19) ≈ log (cid:18) ε (cid:19) , so that | ˜ B ∩ { v = k }|| ˜ B | k ≤ c yields(5.9) | ˜ B ∩ { u j ≤ ε ω ( r j ) }|| ˜ B | ≤ ck ≤ c log log (cid:0) σ (cid:1) , where the constant c log depends only on n, p, s, λ, Λ and the difference of sp/ ( p − α via the definition of ε .We are now in a position to start a suitable iteration to deduce the desired oscil-lation reduction. First, for any i = 0 , , , ... , we define ̺ i = r j +1 + 2 − i r j +1 , ˜ ̺ i := ̺ i + ̺ i +1 , B i = B ̺ i , ˜ B i = B ˜ ̺ i and corresponding cut-off functions φ i ∈ C ∞ ( ˜ B i ) , ≤ φ i ≤ , φ i ≡ B i +1 , and | Dφ i | < c ̺ − i . Furthemore, set k i = (1 + 2 − i ) ε ω ( r j ) , w i := ( k i − u j ) + , and A i = | B i ∩ { u j ≤ k i }|| B i | = | B i ∩ { w i > }|| B i | . The Caccioppoli inequality in (1.8) now yields Z B i Z B i K ( x, y ) | w i ( x ) φ i ( x ) − w i ( y ) φ i ( y ) | p d x d y ≤ c Z B i Z B i K ( x, y )(max { w i ( x ) , w i ( y ) } ) p | φ i ( x ) − φ i ( y ) | p d x d y (5.10) + c Z B i w i ( x ) φ pi ( x ) d x sup y ∈ ˜ B i Z R n \ B i K ( x, y ) w p − i ( x ) d x ! . We can estimate the term on the left below as A pp ∗ i +1 ( k i − k i +1 ) p = 1 | B i +1 | pp ∗ Z B i +1 ∩ { u j ≤ k i +1 } ( k i − k i +1 ) p ∗ φ p ∗ i ( x ) d x ! pp ∗ ≤ | B i +1 | pp ∗ (cid:18)Z B i w p ∗ i ( x ) φ p ∗ i ( x ) d x (cid:19) pp ∗ ≤ c r sp − nj +1 Z B i Z B i K ( x, y ) | w i ( x ) φ i ( x ) − w i ( y ) φ i ( y ) | p d x d y. (5.11)Recalling that | Dφ i | ≤ c i r − j +1 , the first term on the right in (5.10) can be treatedas follows, r spj +1 Z B i Z B i K ( x, y )(max { w i ( x ) , w i ( y ) } ) p | φ i ( x ) − φ i ( y ) | p d x d y ≤ c ip r spj +1 r − pj +1 k pi Z B i ∩ { u j ≤ k i } Z B i | x − y | − p + n + sp d y d x ≤ c ip [ εω ( r j )] p | B i ∩ { u j ≤ k i }| . (5.12)Moreover,(5.13) Z B i w i ( x ) φ pi ( x ) d x ≤ c [ εω ( r j )] | B i ∩ { u j ≤ k i }| OCAL BEHAVIOR OF FRACTIONAL p -MINIMIZERS 23 holds. To tackle the third integral in (5.10), we first have(5.14) r spj +1 sup y ∈ ˜ B i Z R n \ B i K ( x, y ) w p − i ( x ) d x ! ≤ c i ( n + sp ) [Tail( w i ; x , r j +1 )] p − , using inf y ∈ ˜ B i | y − x | ≥ | x − x | inf y ∈ ˜ B i | y − x || x − x | ≥ − i − | x − x | for all x ∈ R n \ B i and the fact that B r j +1 ≡ B j +1 ⊂ B i ⇒ R n \ B i ⊂ R n \ B j +1 . Recalling (5.6) and the facts that w i ≤ εω ( r j ) in B j and w i ≤ | u j | + 2 εω ( r j ) in R n ,we further get[Tail( w i ; x , r j +1 )] p − ≤ cr spj +1 Z B j \ B j +1 w p − i ( x ) | x − x | − n − sp d x + c (cid:18) r j +1 r j (cid:19) sp [Tail( w i ; x , r j )] p − ≤ cε p − ω ( r j ) p − + cσ sp [Tail( u j ; x , r j )] p − ≤ c σ sp − α ( p − ε p − ! [ εω ( r j )] p − ≤ c [ εω ( r j )] p − , by the very definition of ε . Combining the estimates above, we deduce that(5.15) r spj +1 sup y ∈ ˜ B i Z R n \ B i K ( x, y ) w p − i ( x ) d x ! ≤ c i ( n + sp ) [ εω ( r j )] p − . Putting together (5.10), (5.11) (5.12), (5.13) and (5.15), we arrive at A pp ∗ i +1 ( k i − k i +1 ) p ≤ c i ( n + sp + p ) [ εω ( r j )] p A i , which yields A i +1 ≤ c i [ n +(2+ s ) p ] p ∗ /p A βi with β := sp/ ( n − sp ) by the definition of k i ’s. Now, we recall that if we prove thefollowing estimate on A ,(5.16) A = | ˜ B ∩ { u j ≤ εω ( r j ) }|| ˜ B | ≤ c − /β − [ n +(2+ s ) p ] p ∗ / [ pβ ] =: ν ∗ , then we can deduce that A i → i → ∞ . Indeed, the condition (5.16) we can guarantee by (5.9) choosing σ = min { / , exp( − c log /ν ∗ ) } , which then depends only on n, p, s, λ, Λ and the difference of sp/ ( p −
1) and α . Inother words, we have shown thatosc B j +1 u ≤ (1 − ε ) ω ( r j ) = (1 − ε ) (cid:18) r j r j +1 (cid:19) α ω ( r j +1 ) = (1 − ε ) σ − α ω ( r j +1 ) . Taking finally α ∈ (cid:16) , spp − (cid:17) small enough satisfying σ α ≥ − ε = 1 − σ spp − − α , then, clearly, α depends only on n, p, s, λ, Λ andosc B j +1 u ≤ ω ( r j +1 )holds, proving the induction step and finishing the proof. (cid:3) Acknowledgements.
The authors have been supported by the ERC grant 207573 “Vectorial Problems”.The second author has also been supported by Academy of Finland project “Reg-ularity theory for nonlinear parabolic partial differential equations”, and the thirdauthor by PRIN 2010-11 “Calcolo delle Variazioni”. The first and the third authorsare members of Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e leloro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica “F. Severi”(INdAM), whose support is acknowledged.We would like to thank Lorenzo Brasco and Enea Parini for careful reading of apreliminary version of the manuscript. Finally, we would like to thank the refereesfor their useful suggestions, which allowed to improve the manuscript.
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E-mail address , Tuomo Kuusi: [email protected]
E-mail address , Giampiero Palatucci: [email protected] (A. Di Castro)
Dipartimento di Matematica e Informatica, Universit`a degli Studi diParma, Campus - Parco Area delle Scienze 53/A, 43124 Parma, Italy;Dipartimento di Matematica, Universit`a degli Studi di Pisa, Largo B. Pontecorvo 5,56127 Pisa, Italy (G. Palatucci)
Dipartimento di Matematica e Informatica, Universit`a degli Studi diParma, Campus - Parco Area delle Scienze 53/A, 43124 Parma, Italy;SISSA, Via Bonomea 256, 34136 Trieste, Italy (T. Kuusi)(T. Kuusi)