Estimates for Dirichlet-to-Neumann maps as integro-differential operators
aa r X i v : . [ m a t h . A P ] O c t ESTIMATES FOR DIRICHLET-TO-NEUMANN MAPS ASINTEGRO-DIFFERENTIAL OPERATORS
NESTOR GUILLEN, JUN KITAGAWA, AND RUSSELL W. SCHWAB
Abstract.
Some linear integro-differential operators have old and classical representations asthe Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or thegenerator of the boundary process of a reflected diffusion. In this work, we make some extensionsof this theory to the case of a nonlinear
Dirichlet-to-Neumann mapping that is constructed usinga solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data toits normal derivative of the resulting solution. Here we begin the process of giving detailedinformation about the L´evy measures that will result from the integro-differential representationof the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinearDirichlet-to-Neumann mappings. Information about the L´evy measures is important if one hopesto use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings. Introduction, Assumptions, Background
Introduction.
In this work, we explore the precise connection between integro-differentialoperators acting on functions in, e.g. C ,α ( ∂ Ω), for Ω a nice domain in R n +1 , and operators thatare the Dirichlet-to-Neumann mappings (from now on, “D-to-N”) for various elliptic equationsin Ω. We prove estimates on the L´evy measures (explained below) that appear in the integro-differential representation of these D-to-N operators. Our motivating interest is the D-to-N forfully nonlinear elliptic equations (itself, a nonlinear mapping), and the resulting integro-differentialtheory. However, in the course of exploring the nonlinear setting, we noticed the linear theoryseems not to be recorded in any place, except for the case of the Laplacian, where Hsu [26, Section4] gave a complete description for the boundary process of a reflected Brownian motion in asmooth domain. In that sense, this paper can be considered an extension of [26] to the case ofmore general linear and nonlinear equations.The set-up for the D-to-N is as follows. Let Ω be a bounded domain (assumed throughoutfor simplicity, but many adaptations to unbounded domains are possible), let φ ∈ C ,α ( ∂ Ω), andgenerically, we take U φ as the unique solution of ( F ( U φ , x ) = 0 in Ω U φ = φ on ∂ Ω . (1.1)Here F may be any one of the possible operators: Date : Tuesday 6 th March, 2018, ArXiv version 1.2000
Mathematics Subject Classification.
Key words and phrases.
Dirichlet-to-Neumann, integro-differential, nonlocal, elliptic equation, boundary process,fully nonlinear, Levy measures, boundary operators.The authors all acknowledge partial support from the NSF leading to the completion of this work: N. GuillenDMS-1201413 and DMS-1700307; J. Kitagawa DMS-1700094; R. Schwab DMS-1665285. They would like to thankRodrigo Ba˜nuelos and Renming Song for helpful information on background results appearing in Section 2. F ( U, x ) = div( A ( x ) ∇ U ) , with A ∈ C α (Ω) and uniformly elliptic , (1.2) F ( U, x ) = tr( A ( x ) D U ) , with A ∈ C α (Ω) and uniformly elliptic , (1.3) F ( U, x ) = F ( D U, x ) , with F uniformly elliptic with (locally) H¨older coefficients . (1.4)The precise assumptions appear in more detail below. The D-to-N, which we call I , is defined as φ ∂ ν U φ , denoted as I ( φ, x ) := ∂ ν U φ ( x ) , (1.5)where ν ( x ) is the inward normal vector to ∂ Ω at x . In each of these three situations, it is not hardto check (which we do below) that the D-to-N, is not only well defined as a map from C ,α ( ∂ Ω) to C α ( ∂ Ω), but it also enjoys what we call the global comparison property (defined below, Definition1.10). This is the simple fact that the operator, I , preserves ordering between any two functionsthat are globally ordered on ∂ Ω and agree at a point in their domain. The global comparisonproperty of these D-to-N operators is the driving feature behind our results.In the first two of the cases listed in (1.2) and (1.3), F , and hence also I are linear operators.It was proved in the 1960’s, by Bony-Courr`ege-Priouret [5], through linearity and the globalcomparison property, that I must be an integro-differential operator of the form I ( φ, x ) = b ( x ) · ∇ φ ( x ) + p.v. ˆ ∂ Ω ( φ ( h ) − φ ( x )) µ ( x, dh ) , (1.6)for some tangential vector field, b , and a L´evy measure, µ ( x, · ). Recently, two of the authors,in [23], obtained a min-max representation for nonlocal and nonlinear operators that results in aformula similar to (1.6), and in one of our theorems below, we invoke this result to show that I in the nonlinear setting will be a min-max over a family of linear operators of the form (1.6). Wewill record this result precisely in our main results, listed below. We note to the reader that wehave collected various notations in Section 1.2.Our goal is not to re-derive (1.6), but rather to more precisely detail the properties of b and µ .In order to connect I to the recent activity in the theory of linear and nonlinear integro-differentialequations and to exploit some recent results, further properties of the L´evy measures ( µ in (1.6))are required to know which integro-differential results are applicable. This is the main goal of thearticle, and our main results are as follows. We note that we have separated many of the assertionsfor the sake of presentation and that they hold under different assumptions on the regularity of ∂ Ω. Theorems 1.1 and 1.4 have somewhat standard assumptions on ∂ Ω, and Theorem 1.2 requiressignificantly more regularity of ∂ Ω.In the following results, ∂ Ω will be viewed as a Riemannian manifold whose Riemannian metricis induced by the Euclidean inner product on R n +1 . Theorem 1.1 (Linear D-to-N) . Assume that F is as in one of (1.2) or (1.3). If Ω ⊂ R n +1 isbounded and ∂ Ω is of class C with an injectivity radius bounded from below by r > , and I isdefined via (1.1), (1.5), then there exists a vector field, b , and a family of measures parametrizedby x , µ ( x, dh ) , such that for all φ ∈ C ,α ( ∂ Ω) I ( φ, x ) = ( b ( x ) , ∇ φ ( x )) g + ˆ ∂ Ω (cid:16) φ ( h ) − φ ( x ) − B r ( x ) ( h )( ∇ φ ( x ) , exp − x ( h )) g (cid:17) µ ( x, dh ) . (1.7) Furthermore, b and µ satisfy:(i) For all x ∈ ∂ Ω , µ ( x, · ) has a density, µ ( x, dh ) = K ( x, h ) σ ( dh ) ,(ii) There exist universal c > and c ≥ c so that for all x ∈ ∂ Ω , h ∈ ∂ Ω , and x = h , c d ( x, h ) − n − ≤ K ( x, h ) ≤ c d ( x, h ) − n − , -to-N and Integro-Differential Operators 3 (iii) b is bounded.We note that c , c , and the bound for b depend only on the C ,α nature of ∂ Ω in the case of F in (1.2) and only on the C nature of F for (1.3). In addition, if we assume more regularity of ∂ Ω, one can obtain more information about theconstituents of the representation in (1.7).
Theorem 1.2 (H¨older Drift) . If additionally for Ω as above, it is assumed that ∂ Ω is of class C ,then b as in (1.7) is H¨older continuous in x . Remark 1.3.
We note that for Theorem 1.2, we openly admit that assuming ∂ Ω is C is mostlikely more than necessary. However, given that our eventual interest is the hope that some Krylov-Safonov type theorems will be developed for the resulting integro-differential operators, the regularityof b is a low priority. In the context of Krylov-Safonov results, it is the boundedness of b that ismore important, e.g. akin to the results in [47] . Our next result shows that the L´evy measures (away from the singularity) are H¨older continuousin the TV norm. Specifically, it shows that the L´evy measure in (1.7), restricted to the set outsideof a small ball at the singularity, when h = x , enjoys a control that depends on the size of the ballas well as a H¨older fashion in x .We denote by M ( ∂ Ω) the space of signed measures on ∂ Ω, and by k·k
T V the total variationnorm of a signed measure on ∂ Ω. Recall that (see [25, Section 29]) k µ k T V = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) ˆ ∂ Ω φ ( h ) µ ( dh ) (cid:12)(cid:12)(cid:12)(cid:12) : φ ∈ L ∞ ( ∂ Ω) , k φ k L ∞ ( ∂ Ω) ≤ (cid:27) . (1.8) Theorem 1.4 (H¨older in TV Norm) . For a fixed δ > , define µ δ : ∂ Ω → M ( ∂ Ω) by µ δ ( x ) := χ ∂ Ω \ B δ ( x ) ( · ) µ ( x, · ) . Then there exists an α ∈ (0 , such that for δ > sufficiently small, µ δ ∈ C αloc ( ∂ Ω; ( M ( ∂ Ω) , k·k T V )) . More specifically, for each δ there exists a constant C > such that for any x ∈ ∂ Ω and x , x ∈ B δ/ ( x ) it holds that k µ δ ( x ) − µ δ ( x ) k T V ≤ Cδ d ( x , x ) α . Here C depends on universal parameters and the lower bound on the Ricci curvature of ∂ Ω , α arises from the C ,α and C character of ∂ Ω for F respectively in (1.2) and (1.3), while thesmallness required of δ depends only on ∂ Ω . Next, we have the result for the nonlinear version of the D-to-N mapping.
Theorem 1.5 (Nonlinear D-to-N) . If Ω is bounded and ∂ Ω is of class C with an injectivity radiusbounded from below by r > , and I is defined via (1.1), (1.5), using F as in (1.4), then I is amin-max over an appropriate family of operators given by b ij and µ ij , I ( φ, x ) = min i max j { f ij ( x ) + c ij ( x ) φ ( x ) + ( b ij ( x ) , ∇ φ ( x )) g + ˆ ∂ Ω (cid:16) φ ( h ) − φ ( x ) − B r ( x ) ( h )( ∇ φ ( x ) , exp − x ( h )) g (cid:17) µ ij ( x, dh ) } . (1.9) Furthermore,(i) the L´evy measures satisfy, uniformly in i, j , for x, h ∈ ∂ Ω , x = h , N. Guillen, J. Kitagawa, R. Schwab (a) a ring estimate: there exist universal R , C , C , all > , so that for all < r ≤ RC r − ≤ µ ij ( x, B r ( x ) \ B r ( x )) ≤ C r − (b) lower bound: there exists universal R > and η > so that for all h with d ( x, h ) < R and < r < d ( x, h )10 C r η ( d ( x, h )) η +1 ≤ µ ij ( x, B r ( h )) .(ii) b ij is bounded uniformly in i, j .The constants depend on universal parameters and only on the C nature of ∂ Ω . Remark 1.6.
We want to point out to the reader that in both Theorems 1.1 and 1.5, the existenceand boundedness of the b and µ (or f ij , c ij , b ij , µ ij in the min-max) are not new . In the linearcase, this is a result of Bony-Courr`ege-Priouret [5] , and in the nonlinear case by two of the authors [23] . The new part of these results are the properties (i)-(ii) in Theorem 1.1 and (i) in Theorem1.5. Remark 1.7.
The ring estimate in Theorem 1.5 (i-a), although not sufficient for regularity theoryyet, at least shows the the L´evy measures, µ ij , contain the same amount of mass on every ring, B r ( x ) \ B r ( x ) , as does the 1/2-Laplacian. The lower bound in (i-b) at least shows that the L´evymeasures, µ ij are supported everywhere on ∂ Ω , but that possibly they have a scaling that is otherthan the one for surface measure (scaling by the power n ), and we note that one expects η in thissituation to be large (so balls may carry small mass), as opposed to the more regular situationwhere one has η = n . Remark 1.8 ( ∂ Ω ∈ C ) . In both Theorems 1.1 and 1.5, there is an assumption that ∂ Ω shouldbe C . This is a technical assumption arising from the way that the main result in [23] wasproved. There, it is a technical assumption made for simplicity, and so also here it plays the samerole. The more important assumptions arise from results about boundary regularity of solutions ofelliptic equations, in which case, they depend on C ,α or C ingredients, depending upon the typeof equation. Remark 1.9 (Boundedness of Ω) . In all of our results, we have assumed that Ω is bounded. Thisassumption is made purely for simplicity and uniformity, and we note that in many contexts thatthe outcomes of all of the theorems will remain true, provided the supporting results we invokehave modifications to unbounded domains. Some Notation.
Here we collect a list of various notation used in this paper. • We will use capitalized function names, e.g. U (and others), to denote functions defined inthe domain, Ω, and we will use lower case function names, e.g. u (and others), to denotefunctions on the boundary, ∂ Ω. A function solving an equation with prescribed boundarydata would then appear as U u . • Ω is an open bounded domain in R n +1 , that is connected, and with ∂ Ω having an injectivityradius, inj( ∂ Ω), bounded from below by r > • n is the dimension of ∂ Ω, with Ω ⊂ R n +1 . • µ ( x, · ) or µ ij ( x, · ) is a L´evy measure used in the integro-differential representation of I . • d ( x, y ) is the geodesic distance between x and y when x, y ∈ ∂ Ω. • σ is surface measure on ∂ Ω • ν ( x ) is the inward normal vector to ∂ Ω at x ∈ ∂ Ω. • B r ( x ) ⊂ ∂ Ω is a geodesic ball in ∂ Ω and B n +1 r ( x ) ⊂ R n +1 is a Euclidean ball in R n +1 . -to-N and Integro-Differential Operators 5 • The word universal is used for constants that depend only on dimension, ellipticity, ∂ Ω,and the coefficients of F in (1.2)–(1.4). • G ( x, y ) will be the Green’s function for Ω and a linear operator of the form (1.2) or (1.3).1.3. Some Definitions.Definition 1.10.
The global comparison property for I : C ,α ( X ) → C ( X ) requires that for all u, v ∈ C ,α ( X ) such that u ( x ) ≤ v ( x ) for all x ∈ X and such that for some x , u ( x ) = v ( x ) , thenthe operator I satisfies I ( u, x ) ≤ I ( v, x ) . That is to say that I preserves ordering of functionson X at any points where their graphs touch. Definition 1.11.
The second order ( λ, Λ) -Pucci extremal operators are defined as M − and M + ,for a function, U that is second differentiable at x , via M − ( U, x ) = min λ Id ≤ B ≤ ΛId (cid:0) tr( BD U ( x )) (cid:1) and M + ( U, x ) = max λ Id ≤ B ≤ ΛId (cid:0) tr( BD U ( x )) (cid:1) . When { v i } i =1 ,...,n +1 are the eigenvalues of D U ( x ) , an equivalent representation is M − ( U, x ) = Λ X v i ≤ v i + λ X v i > v i and M + ( U, x ) = λ X v i ≤ v i + Λ X v i > v i . Definition 1.12.
We say that F is ( λ, Λ) -uniformly elliptic in the cases (1.2) and (1.3) if λ Id ≤ A ( x ) ≤ ΛId for all x ∈ Ω , and in the case of (1.4) if for all U, V ∈ C (Ω) , M − ( U − V, x ) ≤ F ( D U, x ) − F ( D V, x ) ≤ M + ( U − V, x ) for all x ∈ Ω . We will also require the notion of harmonic measure associated to a linear equation; for detailssee [7, Introduction] for the divergence case and [36, Definition 5.16] for the non-divergence case.
Definition 1.13.
Given linear operators, F (or sometimes L , below), as in (1.2) or (1.3), it iswell known that when φ ∈ C ( ∂ Ω) is prescribed, there exists a unique U φ ∈ C (Ω) that solves (1.1).Thus, for a fixed x , the mapping x U φ ( x ) is well defined, and thanks to the comparison principlefor these equations, is a non-negative linear functional on C ( ∂ Ω) . We take, as a definition, thatfor x fixed, the unique Borel measure that represents this functional to be called the F -Harmonicmeasure (or the L -Harmonic measure), and we denote this measure as ω x . That is to say, ω x , isuniquely characterized by ∀ φ ∈ C ( ∂ Ω) , U φ ( x ) = ˆ ∂ Ω φ ( y ) ω x ( dy ) . Definition 1.14.
Given linear operators, F (or sometimes L , below), as in (1.2) or (1.3), theGreen’s function (see e.g. [36, Section 2] or [41] ) is the unique function such that whenever f isgiven (in an appropriate function space) and U is the unique solution of ( F ( U ) = f in Ω U = 0 on ∂ Ω , then U is uniquely represented as U ( x ) = ˆ Ω f ( y ) G ( x, y ) dy. N. Guillen, J. Kitagawa, R. Schwab
It is a standard result that if ∂ Ω is of class C k , then the tangent bundle T ( ∂ Ω) is of class C k − ,consequently so is the Riemannian metric induced on ∂ Ω by the canonical metric on R n +1 . It canthen be seen that the Riemannian exponential mapping and the geodesic distance squared on ∂ Ωare respectively of class C k − and C k − (see [43, Footnotes, Chapter II, Section 2]).In this paper, we will use the same characterization of a H¨older continuous vector field on ∂ Ωwhich is used in [23]. We record it here for convenience.
Definition 1.15. If V : ∂ Ω → T ( ∂ Ω) is a vector field defined on ∂ Ω , we say V ∈ C αloc ( ∂ Ω) if forany point x ∈ ∂ Ω , there exists an open neighborhood O of x and a constant C > such that | V ( x ) − P y → x V ( y ) | g ≤ Cd ( x, y ) α , ∀ x, y ∈ O , where P y → x is the parallel transport of a vector in T y ( ∂ Ω) to T x ( ∂ Ω) , along the unique geodesicfrom y to x , defined by the Levi-Civita connection of the induced Riemannian metric on ∂ Ω . Background.
The simplest possible case of our map, I , in (1.5) is when Ω = R n +1+ (theupper half space), and F ( U ) = ∆ U . This means that U φ is the harmonic extension of φ , andit is well known that I ( φ ) = − ( − ∆) / φ . This corresponds to the generator of the boundaryprocess, after a time rescaling, of a reflected Brownian motion in R n +1+ , recording the locations ofthe process restricted to the plane R n × { } . This well known fact was generalized to boundeddomains, Ω, as above, by Hsu in [26], which characterizes the generator of this boundary processas I , and gives some properties, such as those in Theorem 1.1, above. This is in the context ofthe well-known relationship between D-to-N mappings and generators for boundary processes ofgeneral reflected diffusions (rescaled using their local time), and some good references are e.g. [44,Sec. 8] and [29, Chp. IV, Sec. 7]. Thus, one can see Theorem 1.1 as a generalization of [26] tomore general diffusion processes with H¨older diffusion coefficients.There is, however, a different reason for our goals in this paper beyond simply to extend [26]to more general linear and nonlinear settings. This is the desire to give a more precise linkbetween D-to-N mappings and integro-differential equations, with the hopes of leveraging newresults for integro-differential operators. Developments of approximately the last 20 years haveled to good understanding of the regularity for solutions of equations that involve linear and fullynonlinear integro-differential operators similar to (1.6)– at least in the case that ∂ Ω = R n . Thus,it seems reasonable to further pursue the link between the integro-differential theory and D-to-N mappings, with the hope that recent results in the integro-differential theory could possiblylead to new understanding or results involving Neumann problems. Two of the developmentsin the integro-differential world that could be of use are, broadly speaking: regularity resultsthat use only the roughest bounds on coefficients and L´evy measures– we can call these Krylov-Safonov type estimates (we mention some specific results in the next paragraphs); and the recentresult of two of the authors that shows that under certain conditions (established below) that theD-to-N mapping for fully nonlinear equations can be represented as a min-max over linear integro-differential operators [23]. In order to connect these two developments, one must, of course, gainfurther information about the µ (or µ ij ) that appear in Theorems 1.1 and 1.5.In its simplest presentation, a Krylov-Safonov result basically says that for a linear operatorsuch as in (1.6), the solutions, say u , of Lu = f in B satisfy the H¨older estimate, for a universal C ,[ u ] C α ( B / ) ≤ C ( k u k L ∞ ( R n ) + k f k L ∞ ( B ) ) . (1.10) -to-N and Integro-Differential Operators 7 This has been pursued under various lists of assumptions from many various authors, and welist some explicitly below. This estimate may seem simple, but it’s importance as one of thefew compactness tools for non-divergence form equations cannot be overstated. This result wasa cornerstone of the local, second order, elliptic theory, dating back the the original work ofKrylov-Safonov [38].In recent years, Krylov-Safonov type results have been obtained for nonlocal operators like (1.6)by many authors, and here we mention some of the results in this direction, and we indicate thatthis list is by no means complete. Bass-Levin [3] proved (1.10) for the class where (for α ∈ (0 , b ≡ , µ ( x, dh ) = k ( x, h ) dh, k ( x, − h ) = k ( x, h ) , and λ | h | n + α ≤ k ( x, h ) ≤ Λ | h | n + α . (1.11)Bass-Kassmann [2], Song-Vondracek [49], and subsequently Silvestre [46] (also including slightlymore general k ) extended this to the same setting, except that variable exponents, α ( x ), could beallowed: λ | h | n + α ( x ) ≤ k ( x, h ) ≤ Λ | h | n + α ( x ) , for α ( x ) ∈ (0 , − c ) , c > . Finally, along this line of attack, with similar assumptions as in (1.11), Caffarelli-Silvestre [8]obtained (1.10) for those kernels that satisfy k ( x, − h ) = k ( x, h ) and (2 − α ) λ | h | n + α ≤ k ( x, h ) ≤ (2 − α )Λ | h | n + α , and furthermore, their proof obtained the result (1.10) in a way that is independent of α closeto 2 (the assumption that includes the factor (2 − α ) is consistent with the α/ α → k ( x, − h ) = k ( x, h )in (1.11); (ii) relaxing the lower bounds, λ | h | − d − α ≤ k ( x, h ), in (1.11); and (iii) extending thetheory to include parabolic equations. Results that have relaxed requirements on the symmetryof k include: Chang Lara [11], Chang Lara - D´avila [13] and [14], Schwab-Silvestre [45]. Resultsthat have relaxed requirements on the lower bounds on k include: Bjorland-Caffarelli-Figalli [4],Guillen-Schwab [22], Kassmann-Mimica [33], Kassmann-Rang-Schwab [35], and [45]. Results thathave extended the above to the parabolic setting include: [12], [14], and [45]. Finally, we notethat there is an extension that is completely separate from all of the others listed here in that itobtains Krylov-Safonov estimates in the situation that the exponent, α , in (1.11) is allowed to godown to α = 0 as well as allows for scaling laws that are more general than (1.11); this is the workof Kassmann-Mimica [34], followed up by the work of Kim-Kim-Lee [37] .There are many uses for the D-to-N, and we would like to point out the work of Hu-Nicholls[27], where they study the dependence of the D-to-N on changes to the domain, Ω (for a slightlydifferent family of equations). There are also many useful references for related issues in [27].We conclude this section by mentioning that only in the simplest setting that Ω = R n +1+ and F is given by (1.2) or (1.3) will some of the above mentioned results involving non-symmentric k apply to the operator I that results from Theorem 1.1. In the case that F is nonlinear or in allcases when ∂ Ω is not flat, none of the above mentioned results apply to I . This suggests room formore study on this issue, and we briefly elaborate on this in Section 7. N. Guillen, J. Kitagawa, R. Schwab Some Useful Tools For Boundary Behavior
In this section, we collect some various results that will be useful later on. The followingproposition is about the boundary behavior of the Green’s function for C ,α domains. The upperbound is a special case of the estimates for equations with H¨older coefficients in nice domains thatcan be found in Gr¨uter-Widman [21]. The lower bound is a consequence of the Harnack inequalityand is outlined in the proof of the main result of Zhao [50]. Proposition 2.1 (Constant Coefficient) . Assume that ∂ Ω is a C ,α boundary. For the constantcoefficient operator, i.e. Lu = ∆ U , it holds that for the Green’s function, G ( x, y ) , for all x, y ∈ Ω c d ( x ) d ( y ) | x − y | n +1 ≤ G ( x, y ) ≤ c d ( x ) d ( y ) | x − y | n +1 . Here we use d ( x ) = d ( x, ∂ Ω) . ( d ( x, ∂ Ω) = inf y ∈ ∂ Ω | x − y | , and recall, Ω ⊂ R n +1 ) After taking normal derivatives of the Green’s function, this gives in [50],
Proposition 2.2 (Poisson Kernel Constant Coefficients, [50]) . Assume that ∂ Ω is a C ,α bound-ary. For the constant coefficient operator, i.e. LU = ∆ U , it holds that for the Poisson kernel, P ( x, y ) , for all x ∈ Ω and z ∈ ∂ Ω c d ( x ) | x − z | n +1 ≤ P ( x, z ) ≤ c d ( x ) | x − z | n +1 . (Recall, Ω ⊂ R n +1 ) It turns out that the same behavior was extended to variable coefficients by respectively Cho[15] and Hueber-Sieveking [28]. We record this here
Proposition 2.3 (Variable Coefficients) . (a) (Hueber-Sieveking [28] ) Assume that LU = tr( A ( x ) D U ( x )) + B ( x ) · ∇ U ( x ) + C ( x ) U ( x ) , with H¨older coefficients and that ∂ Ω is C , . Then Proposition 2.1 remains true.(b) (Cho [15] ) Assume that LU = div( A ( x ) ∇ U ( x )) , with H¨older coefficients, and that ∂ Ω is C ,α . Then Proposition 2.1 remains true. We note that Cho [15] proves the estimate for the Heat kernel, but the result for the ellipticproblem follows from the identity G ( x, y ) = ˆ ∞ p ( x, y, t ) dt, where p ( x, y, t ) is the heat kernel, or transition density function for the corresponding killed processin Ω (i.e. the fundamental solution of the heat equation with zero boundary data).Finally, we state here a relationship between the F -Harmonic measure and the Green’s func-tion for a linear equation. We note that the Green’s function for non-divergence equations arewell known not to be well behaved pointwise; however, in light of the fact that we are dealingwith equations with H¨older coefficients, this is a situation where the Green’s function is definedpointwise, furthermore we record the actual result we use in the next lemma. Proposition 2.4 (Harmonic Measure - Green Function estimates) . Let { ω x } x ∈ Ω be the F -harmonicmeasure where F is defined by (1.2) or (1.3) , and G be the Green’s function for F on Ω . -to-N and Integro-Differential Operators 9 Then there are universal constants ρ , C , C > and s > such that for any ρ ∈ (0 , ρ ) , x ∈ ∂ Ω , and y ∈ Ω \ B s ρ ( x ) , for the divergence equation (1.2) it holds C ρ n − G ( y, x + ρν ( x )) ≤ ω y ( ∂ Ω ∩ B ρ ( x )) ≤ C ρ n − G ( y, x + ρν ( x )) , and for the non-divergence equation (1.3) it holds that C ρ ˆ ˜ B ρ G ( y, z ) dz ≤ ω y ( ∂ Ω ∩ B ρ ( x )) ≤ C ρ ˆ B n +1 ρ ( x ) ∩ Ω G ( y, x ) dz where ˜ B ρ = B n +1 ( x + ν ( x )) Proof.
The divergence case (1.2) is an immediate consequence of [7, Lemma 2.2].For the non-divergence case, both bounds appear in the proof of [36, Lemma 5.18]. We alsouse the lower bound explicitly as a crucial step in Section 5, and so the proof of that inequalityappears in the proof of Lemma 5.6, below. (cid:3)
Lemma 2.5 (Comparison of intrinsic and extrinsic annuli) . There exists an ǫ > such that forany r ∈ (0 , ǫ ) and x ∈ ∂ Ω , ( B n +1(7 / r ( x ) \ B n +1(5 / r ( x )) ∩ ∂ Ω ⊂ ( B r ( x ) \ B r ( x )) ⊂ ( B n +1(9 / r ( x ) \ B n +1(3 / r ( x )) ∩ ∂ Ω . Proof.
By definition of geodesic distance, it is clear that for any h , x ∈ ∂ Ω, | x − h | ≤ d ( x , h ) . Now since ∂ Ω is C , for any x ∈ ∂ Ω there exists ǫ x > ρ x ∈ C ( B nǫ x ( x )) with (after a rotationof coordinates) B n +1 ǫ x ( x ) ∩ Ω = { ( y ′ , y n +1 ) ∈ B n +1 ǫ x ( x ) : ρ x ( y ′ ) < y n +1 } ,ρ x ( x ) = 0 , ∇ ρ x ( x ) = 0 , q k ρ x k C ( B nǫx ( x )) ≤ . By compactness of ∂ Ω we can see ǫ := inf x ∈ ∂ Ω ǫ x >
0. Now let x ∈ ∂ Ω, rotate coordinatesto identify R n with T x ( ∂ Ω). We claim that the projection π R n ( B ǫ ( x )) into R n is contained in B nǫ ( x ). Suppose this is not the case, so h = ( h ′ , h n +1 ) ∈ B ǫ ( x ) but h ′ B nǫ ( x ). We can take alength minimizing C curve γ : [0 , → ∂ Ω connecting x to h , which we assume constant speed,and let t := inf { t ∈ [0 , | π R n ( γ ( t )) B nǫ ( x ) } >
0. Then we calculate ǫ > d ( x , h ) ≥ ˆ t | ˙ γ ( t ) | dt ≥ ˆ t | π R n ( ˙ γ ( t )) | dt ≥ ǫ , a contradiction.Thus if h ∈ B ǫ ( x ), we can write h := ( y ′ , ρ x ( y ′ )) for some y ′ ∈ B nǫ ( x ), then d ( x , h ) ≤ ˆ (cid:12)(cid:12) ( y ′ , ( ∇ ρ x ( ty ′ ) · y ′ )) (cid:12)(cid:12) dt ≤ r | y ′ | + k ρ x k C ( B nǫ ( x )) | y ′ | ≤ (cid:12)(cid:12) y ′ (cid:12)(cid:12) ≤ | x − h | , and the claimed inclusions immediately follow. (cid:3) Well-Posedness and Lipschitz Nature of the D-to-N
Here we record the relatively straightforward facts that I defined via (1.1) and (1.5) is in factwell defined and a Lipschitz mapping C ,α → C α in each of the three instances (1.2), (1.3), (1.4). Lemma 3.1.
If the equation (1.1) satisfies the assumptions(i) φ ∈ C ,α ( ∂ Ω) = ⇒ U φ ∈ C ,α ′ (Ω) for some < α ′ < α (Regularity)(ii) φ ≤ ψ on ∂ Ω = ⇒ U φ ≤ U ψ in Ω (Comparison)(iii) φ ∈ C ( ∂ Ω) = ⇒ U φ exists and is unique in the (weak, viscosity, strong, or classical sense),then the D-to-N mapping, I , defined in (1.1) and (1.5) is well defined and has the global comparisonproperty over C ,α ( ∂ Ω) .Proof. First of all, the assumption of existence and uniqueness of U φ , combined with the assump-tion (Regularity) at least show that I is well defined as a map from C ,α ( ∂ Ω) to C α ′ ( ∂ Ω). The onlything to check is the comparison property. However, I inherits this directly from the assumption(Comparison) that is made on F . Indeed, let u , v , and x ∈ ∂ Ω be given such that u ≤ v on ∂ Ωand that u ( x ) = v ( x ). Let ν ( x ) be the inward normal vector at x and let h > U u ( x + hν ( x )) − U u ( x ) ≤ U v ( x + hν ( x )) − U v ( x ) , and thus since ∂ ν U u and ∂ ν U v exist by (Regularity), we conclude ∂ ν U u ( x ) ≤ ∂ ν U v ( x ) . (cid:3) Just for completeness, we include a list of results which establish the assumptions of comparisonand regularity made in Lemma 3.1 for each of the cases of F in (1.2)–(1.4). Lemma 3.2. If F is given as (1.2), (1.3), or (1.4), then the equation (1.1) satisfies the assump-tions of (regularity), (comparison), (existence/uniqueness) listed in Lemma 3.1.Proof of Lemma 3.2. In the case of (1.2), weak solutions are defined via the bilinear form, B ( u, v ) = ˆ Ω ∇ u ( x ) · A ( x ) ∇ v ( x ) dx, and the establishment of uniqueness, comparison, and regularity under the assumption that A ∈ C α (Ω) can be found in [20, Chp 8].In the case of (1.3), “weak” solutions can be understood as either strong solutions e.g. [20, Chp9] or viscosity solutions e.g. [17] (both cases are equivalent for this equation and these assump-tions). Note, in this case, U φ is actually C ,αloc (Ω), but not in the whole of Ω as we only assume φ ∈ C ,α ( ∂ Ω). The assumptions that A ∈ C α (Ω) and is uniformly elliptic imply uniqueness,comparison, and regularity, and can be found in [20, Chp 9], among other sources.Finally, in the case of (1.4), the “locally H¨older coefficients” assumption means that for allsymmetric matrices, P , | F ( P, x ) − F ( P, y ) | ≤ C | x − y | α (1 + k P k ) , and for simplicity we can assume that F (0 , x ) ≡
0. The notion of weak solution is viscositysolutions, e.g. [17]. We refer to [48, Theorem 1.4] for the validity of the C ,α estimates in(regularity), and to [30, Theorem III.1] for the validity of the comparison result, which in thiscontext also gives the uniqueness of the viscosity solution. (cid:3) -to-N and Integro-Differential Operators 11 Just as in the case of second order elliptic equations, it will be useful to understand whichoperators govern the ellipticity class for the D-to-N, I , in the context of F in (1.4) (i.e. theanalogous objects to the Pucci operators for second order equations that appear in Definition1.11). It turns out that a convenient choice of these extremal operators are the D-to-N operatorsfor the second order extremal operators. The following observation is copied from [24, Lemma3.3]: Lemma 3.3.
In (1.1), take F to be respectively M − and M + which are in Definition (1.11),and take respectively U − φ and U + φ to be the corresponding solutions of (1.1). Define the boundaryextremal operators as M − ( φ, x ) := ∂ ν U − φ ( x ) and M + ( φ, x ) := ∂ ν U + φ ( x ) . (3.1) Then M ± are extremal operators for I in the sense that for all u, v ∈ C ,α ( ∂ Ω) and for all x ∈ ∂ Ω M − ( u − v, x ) ≤ I ( u, x ) − I ( v, x ) ≤ M + ( u − v, x ) . (3.2) Proof.
Here F is a fixed uniformly elliptic operator from (1.4). For ease of presentation, we recordthe two different equations that are being used here: ( F ( D U, x ) = 0 in Ω U = φ on ∂ Ω . (3.3)and ( M + ( U, x ) = 0 in Ω U = φ on ∂ Ω . (3.4)Let U u and U v be the unique solutions of (3.3) with respectively boundary data given by φ = u and φ = v . We will just prove the upper bound, and the lower bound follows analogously.We note that since U u and U v are respectively a viscosity sub and super solution of (3.4), thenit follows that U u − U v is a viscosity subsolution of0 ≤ M + ( U u − U v ) . Hence, if U +( u − v ) is the solution to (3.4) with φ = u − v , since U +( u − v ) and U u − U v have the sameboundary data, the comparison of sub and super solutions for (3.4) shows that U u − U v ≤ U +( u − v ) in Ω and U u − U v = u − v = U +( u − v ) on ∂ Ω . Hence ∂ ν U u − ∂ ν U v ≤ ∂ ν U +( u − v ) = M + ( u − v ) , which concludes the lemma. (cid:3) Lemma 3.4.
In all cases of (1.2), (1.3), (1.4), there exists some choice of α ′ with < α ′ < α ,so the D-to-N, I , is a Lipschitz mapping of C ,α ( ∂ Ω) → C α ′ ( ∂ Ω) . I also satisfies the extraassumption in [23, Theorem 1.6 - (1.3)] , which requires ∀ u, v ∈ C ,α ( ∂ Ω) , kI ( u ) − I ( v ) k L ∞ ( B r ) ≤ C (cid:16) k u − v k C ,α ( B r ) + ω ( r ) k u − v k L ∞ ( ∂ Ω) (cid:17) , (3.5) and ω ( r ) → as r → ∞ . Proof.
First, we remark on the special assumption (1.3) in [23, Theorem 1.6], which we listed hereas (3.5). In this context, we simply require that the normal derivative of the solution, U u , in B r iscontrolled by k u k C ,α ( B r ) , which is a standard type of estimate for boundary regularity. We recallthat we are assuming for simplicity that Ω is bounded. Hence, (3.5) is trivial once the Lipschitzcharacter of I is established, as we can just take ω ( r ) ≡ r > diam(Ω).The Lipschitz nature of I follows from the global (up to the boundary) C ,α ′ regularity theoryfor (1.1). Let u, v ∈ C ,α ( ∂ Ω). In the two linear cases, (1.2) and (1.3), we note that (with apologiesfor the triviality) I ( u ) − I ( v ) = I ( u − v ) , and in the nonlinear case (1.4) that we will invoke the extremal inequalities (3.2), which meanswe will be utilizing boundary regularity theory for M − ( U u − v , x ) = 0 , and M + ( U u − v , x ) = 0 in Ω . (3.6)For the divergence case, (1.2), one reference is [20, Theorem 8.33], and for the non-divergencecase, (1.3), the regularity is a straightforward consequence for the boundary oscillation reductionof the quantity U ( x ) /d ( x, ∂ Ω) that can be found in [20, Theorem 9.31]. For the nonlinear case(1.4) one reference is [48, Theorem 1.1], applied to each of the equations in (3.6). All of theseresults imply that for a universal C k U φ k C ,α ′ (Ω) ≤ C (cid:16) k U φ k L ∞ (Ω) + k φ k C ,α ( ∂ Ω) (cid:17) , (3.7)and when combined with the maximum principle, | U φ | ≤ k φ k L ∞ ( ∂ Ω) , we see that kI ( φ, · ) k C α ′ ( ∂ Ω) ≤ C k φ k C ,α ( ∂ Ω) . Hence, applying this in each of our cases to φ = u − v , we obtain the Lipschitz bound. (cid:3) Linear equations with H¨older coefficients– Proofs of Theorems 1.1, 1.2, 1.4
In this section, we include the proofs of Theorems 1.1, 1.2, 1.4. We note that the existence of b and µ , the validity of (1.7), boundedness of b are all a direct result of [23, Theorem 1.6 andProposition 1.7].4.1. Density and bounds for µ (Proof of Theorem 1.1). Proof of Theorem 1.1.
Fix x ∈ ∂ Ω, we show that µ ( x, · ) is absolutely continuous with respect tosurface measure, σ , on ∂ Ω on ∂ Ω \ { x } . This will be done by showing absolute continuity on theset ∂ Ω \ { B r ( x ) } for any arbitrary r >
0, then we can exhaust ∂ Ω \ { x } by a union of such sets.Thus fix r > E ⊂ ∂ Ω \ { B r ( x ) } with σ ( E ) = 0.Fix δ >
0, then we find a countable cover { B ( x j , r j ) } ∞ j =1 of E by open geodesic balls such that P ∞ j =1 r nj < δ ; let us write B j := B ( x j , r j ) for brevity. Now let φ ∈ C ( ∂ Ω) be any function suchthat 0 ≤ φ ≤ S ∞ j =1 B j . If δ is sufficiently small compared to r , we will have φ ≡ B r/ ( x ) thus ∇ φ ( x ) = 0, so in (1.7)we have I ( φ, x ) = ˆ ∂ Ω \ B r/ ( x ) φ ( y ) µ ( x, dy ) . -to-N and Integro-Differential Operators 13 Let { ω x } x ∈ Ω be the F -harmonic measure for F given by (1.2) (see Definition 1.13), then recall U φ ( x ) = ˆ ∂ Ω φ ( y ) ω x ( dy )for any x ∈ Ω. Now if s > j by Proposition 2.4 we have ω x + sν ( x ) ( ∂ Ω ∩ B j ) ≤ ( Cr n − j G ( x + sν ( x ) , x j + r j ν ( x j )) for (1 . C r j ´ B n +1 rj ( x ) ∩ Ω G ( x + sν ( x ) , z ) dz for (1 . . where G is the Green’s function and C depends only on ∂ Ω and the ellipticity of the equation.Thus we have the estimate U φ ( x + sν ( x )) = ˆ ∂ Ω φ ( y ) ω x + sν ( x ) ( dy ) ≤ ∞ X j =1 ω x + sν ( x ) ( ∂ Ω ∩ B j ) ≤ C ∞ X j =1 ( r n − j G ( x + sν ( x ) , x j + r j ν ( x j )) for (1 . r j ´ B n +1 rj ( x ) ∩ Ω G ( x + sν ( x ) , z ) dz for (1 . . ≤ C ∞ X j =1 r n − j sr j | x + sν ( x ) − ( x j + r j ν ( x j )) | n +1 for (1 . r j · sr n +2 j | x + sν ( x ) − ( x j + r j ν ( x j )) | n +1 for (1 .
3) (4.1) ≤ C r s ∞ X j =1 r nj < C r sδ where we have used Proposition 2.3 to obtain the second to final inequality and C r is some constantdepending on n , r , ellipticity, and ∂ Ω (but independent of δ and φ ). Thus ˆ ∂ Ω \ B r/ ( x ) φ ( y ) µ ( x, dy ) = ∂ ν U φ ( x ) = lim s → U φ ( x + sν ( x )) − φ ( x ) s ≤ C r δ. Since { B j } covers E , we can take a sequence of C ( ∂ Ω) functions E ≤ φ k ≤ S ∞ j =1 B j decreasingpointwise to E to obtain µ ( x, E ) ≤ C r δ , and since δ was arbitrary this yields µ ( x, E ) = 0.By the above, we can write µ ( x, dy ) = K ( x, y ) σ ( dy ) for some density K when restricted to ∂ Ω \ { x } . Fix y = x in ∂ Ω and 0 < r < | x − y | , and this time let φ rl and φ ru ∈ C ( ∂ Ω) be suchthat 0 ≤ φ rl ≤ B r ( y ) ≤ φ ru with φ rl ≡ B r/ ( y ) φ rl ≡ ∂ Ω \ B r ( x ) φ ru ≡ B r/ ( x ) φ ru ≡ ∂ Ω \ B r ( x ) . Following similar calculations as before, and invoking the same split argument for the divergence/non-divergence setting in (4.1), we have U φ rl ( x + sν ( x )) = ˆ ∂ Ω φ rl ( z ) ω x + sν ( x ) ( dz ) ≤ ω x + sν ( x ) ( ∂ Ω ∩ B r ( y )) ≤ C sr n | x + sν ( x ) − ( y + rν ( y )) | n +1 . Since again φ rl ≡ x , we have I ( φ rl , x ) = ´ ∂ Ω ∩ B r ( x ) φ rl ( y ) µ ( x, dy ), hence taking the limit inthe difference quotient we have µ ( x, ∂ Ω ∩ B r ( x )) σ ( ∂ Ω ∩ B r ( x )) ≤ σ ( ∂ Ω ∩ B r ( x )) ˆ ∂ Ω ∩ B r ( x ) φ rl ( y ) µ ( x, dy )= ∂ ν U φ rl ( x ) σ ( ∂ Ω ∩ B r ( x )) ≤ C | x − ( y + rν ( y )) | n +1 . a similar calculation utilizing φ ru yields µ ( x, ∂ Ω ∩ B r ( x )) σ ( ∂ Ω ∩ B r ( x )) ≥ C | x − ( y + rν ( y )) | n +1 . By the Lebesgue differentiation theorem, for σ -a.e. y ∈ ∂ Ω we have C | x − y | n +1 ≤ K ( x, y ) = lim r → µ ( x, ∂ Ω ∩ B r ( x )) σ ( ∂ Ω ∩ B r ( x )) ≤ C | x − y | n +1 . (cid:3) H¨older continuity of the coefficients of I (Proof of Theorem 1.2). Before embarkingon the proof of Theorem 1.1, we make some background observations. Recall 2 r > ∂ Ω. For this portion we assume ∂ Ω to be a C surface, this means the tangent bundle T ( ∂ Ω) is a C manifold. Then the restriction of theEuclidean metric from R n +1 to ∂ Ω is also C , and the exponential mapping exp x based at anypoint x ∈ ∂ Ω is C (the same holds for its inverse in its domain of definition). In particular thegeodesic distance squared will be C on B r ( x ) × B r ( x ), meaning that the second derivativeinvolving the mapping D (exp − p ) | h leading to the estimate (4.13) below is justified. Finally, recallDefinition 1.15 for the H¨older continuity of a vector field on ∂ Ω. Proof of Theorem 1.2.
Fix x , y ∈ ∂ Ω which will be taken so d ( x , y ) is smaller than someuniversal constant, that is yet to be determined. For ease of notation let us write d := d ( x , y ) , and we tacitly assume d ≤ min { , r } . Also fix a unit length v ∈ T x ( ∂ Ω), and let φ be a C function on ∂ Ω such that for h ∈ B r ( x ) we have φ ( h ) = ( v, exp − x ( h )) g . Computing using normal coordinates centered at x we easily see ∇ φ ( x ) = v , and in particular φ ( h ) = ( ∇ φ ( x ) , exp − x ( h )) g on B r ( x ). Also let η ∈ C ∞ ( R ) be such that 0 ≤ η ≤ η ≡ , r ], and η ≡ r + d α , ∞ ) for some α ∈ (0 ,
1) which will be determined later; we will alsoassume that r + d α is less than the injectivity radius of ∂ Ω, and so d α ≤ r will suffice. We -to-N and Integro-Differential Operators 15 also define η x , η y by η ( d ( x , · )) and η ( d ( y , · )) respectively, both of which can be seen to be C .Then I ( η x φ, x )= ( b ( x ) , ∇ ( η x φ )( x )) g + ˆ ∂ Ω (cid:16) ( η x φ )( h ) − ( η x φ )( x ) − B r ( x ) ( h )( ∇ ( η x φ )( x ) , exp − x ( h )) g (cid:17) µ ( x , dh )= ( b ( x ) , ∇ φ ( x )) g + ˆ ∂ Ω (cid:16) η x ( h ) φ ( h ) − B r ( x ) ( h )( ∇ φ ( x ) , exp − x ( h )) g (cid:17) µ ( x , dh )= ( b ( x ) , v ) g + ˆ B r dα ( x ) \ B r ( x ) ( η x ( h ) −
1) ( v, exp − x ( h )) g µ ( x , dh ) . Now for points x , y ∈ ∂ Ω such that y ∈ B r ( x ) let P x → y denote parallel transport of a tangentvector from x to y along the minimal geodesic connecting x to y . In a manner similar to theconstruction of φ , we take ψ to be a C function on ∂ Ω such that ψ ( h ) = ( P x → y v, exp − y ( h ))for h ∈ B r ( y ).Then a similar calculation as above yields I ( η y ψ, y ) = ( b ( y ) , P x → y v ) g + ˆ B r dα ( y ) \ B r ( y ) η y ( h )( P x → y v, exp − y ( h )) g µ ( y , dh ) . Thus, using the fact that parallel transport preserves inner product, | ( b ( x ) − P y → x b ( y ) , v ) g | = | ( b ( x ) , v ) g − ( b ( y ) , P x → y v ) g | . Thus, using the triangle inequality, we can continue the previous as: | ( b ( x ) − P y → x b ( y ) , v ) g | ≤ |I ( η x φ, x ) − I ( η y ψ, y ) | (4.2)+ | ˆ B r dα ( x ) \ B r ( x ) ( η x ( h ) −
1) ( v, exp − x ( h )) g µ ( x , dh ) | + | ˆ B r dα ( y ) \ B r ( y ) ( η y ( h ) −
1) ( P x → y v, exp − y ( h )) g µ ( y , dh ) | =: I + II + III. (4.3)Now for the terms II and III , we calculate using Theorem 1.1 part (ii), II ≤ ˆ B r dα ( x ) \ B r ( x ) (cid:12)(cid:12) exp − x ( h ) (cid:12)(cid:12) g µ ( x , dh ) ≤ g ( ∂ Ω) ˆ B r dα ( x ) \ B r ( x ) d ( x , h ) − n − σ ( dh ) ≤ g ( ∂ Ω) r − n − σ ( B r + d α ( x ) \ B r ( x )) ≤ Cd α (4.4)for some universal C >
0. We obtain the estimate for
III in the same.The remainder of the proof is to estimate the term I . Since η x φ and η y ψ are C functionson ∂ Ω, we can use the results mentioned in the discussion preceding and following (3.7). That is, there is some β ∈ (0 , < β ′ < β , and a universal C >
0, so that I ≤ (cid:12)(cid:12)(cid:12) ∂ ν U η x φ ( x ) − ∂ ν U η y ψ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ∂ ν U η y ψ ( x ) − ∂ ν U η y ψ ( y ) (cid:12)(cid:12)(cid:12) ≤ |I ( η x φ, x ) − I ( η y ψ, x ) | + C | x − y | β ′ [ ∇ U η y ψ ] C β ′ (Ω) ≤ |I ( η x φ, x ) − I ( η y ψ, x ) | + Cd β ′ k η y ψ k C ,β ( ∂ Ω) . (4.5)It is easy to see that k η y ψ k C ,β ( ∂ Ω) is bounded by a universal constant times d − α , hence d β ′ k η y ψ k C ,β ( ∂ Ω) ≤ Cd β ′ − α . (4.6)To deal with the first term, take ρ > r also to be determined later, and let˜ η ∈ C ∞ ( R ) with 0 ≤ ˜ η ≤
1, ˜ η ≡ − ρ, ρ ] and ˜ η ≡ ρ, ∞ ), and define ˜ φ := ˜ η ( d ( x , · )) ∈ C ,β ( ∂ Ω). It is easy to see that both k ˜ φ k C ( ∂ Ω) and k − ˜ φ k C ( ∂ Ω) are bounded by a universalconstant times ρ − . We then apply [23, Lemma 4.15 (4.7)] and use Lemma 3.4 to see that (afterpossibly making a smaller choice for β ), |I ( η x φ, x ) − I ( η y ψ, x ) | ≤ C ( k (1 − ˜ φ )( η x φ − η y ψ ) k C ,β ( ∂ Ω) + k ˜ φ k C ,β ( ∂ Ω) k η x φ − η y ψ k L ∞ (spt( ˜ φ )) ) ≤ C ( k (1 − ˜ φ )( φ − ψ ) k C ,β ( ∂ Ω) + ρ − − β k η x φ − η y ψ k L ∞ ( ∂ Ω \ B ρ ( x )) ) . (4.7)Now note for any h ∈ ∂ Ω, | η x ( h ) φ ( h ) − η y ( h ) ψ ( h ) | ≤ | η x ( h ) | | φ ( h ) − ψ ( h ) | + | η x ( h ) ψ ( h ) − η y ( h ) ψ ( h ) |≤ | η x ( h ) | | φ ( h ) − ψ ( h ) | + diam g ( ∂ Ω) | η ( d ( x , h )) − η ( d ( y , h )) |≤ | η x ( h ) | | φ ( h ) − ψ ( h ) | + C k η k C ( R ) | d ( x , h ) − d ( y , h ) |≤ | η x ( h ) | | φ ( h ) − ψ ( h ) | + Cd − α . (4.8)The first term in the last line above is zero unless d ( x , h ) ≤ r + d α . For such h we find | η x ( h ) φ ( h ) − η x ( h ) ψ ( h ) | ≤ (cid:12)(cid:12) ( v, exp − x ( h ) − P y → x exp − y ( h )) g (cid:12)(cid:12) ≤ (cid:12)(cid:12) exp − x ( h ) − P y → x exp − y ( h ) (cid:12)(cid:12) g = 12 (cid:12)(cid:12)(cid:0) ∇ x d ( x, h ) | x = x − P y → x ∇ x d ( x, h ) | x = y (cid:1)(cid:12)(cid:12) g ≤
12 sup x ∈ ∂ Ω ,h ∈ B r ( x ) (cid:12)(cid:12) Hess x d ( · , h ) (cid:12)(cid:12) g d ( x , y ) ≤ Cd , where to obtain the third line above we have used [32, Theorem 5.6.1 (5.6.4)]. Thus if we take ρ := d α / (1+ β )0 for α ∈ (0 ,
1) to be determined , (4.9)by (4.8) we have ρ − − β k η x φ − η y ψ k L ∞ ( ∂ Ω \ B ρ ( x ) ≤ Cd − α − α . (4.10)Next we turn to the term k (1 − ˜ φ )( φ − ψ ) k C ,β ( B ρ ( x )) . First, k (1 − ˜ φ )( φ − ψ ) k C ( B ρ ( x )) ≤ k φ − ψ k C ( B ρ ( x )) ≤ Cd by the same argument as above. -to-N and Integro-Differential Operators 17 Next fix any h ∈ B ρ ( x ), w ∈ T h ( ∂ Ω) and define for t ∈ [0 ,
1] and s near zero, γ ( s, t ) : = exp y ( t (exp − y ( h ) + s [ D (exp − y ) | h ( w )])) ,J ( t ) : = ∂∂s | s =0 γ ( s, t ) , then J is a Jacobi field along the geodesic from y to h with J (0) = 0 and ˙ J = D (exp − y ) | h ( w )(see [42, Sec 6.1.4]). Then we calculate two different ways, ∂∂s | s =0 ψ ( γ ( s, ∂∂s | s =0 ( P x → y v, exp − y ( h ) + s [ D (exp − y ) | h ( w )]) g = ( P x → y v, D (exp − y ) | h ( w )) g = ( v, P y → x [ D (exp − y ) | h ( w )]) g ,∂∂s | s =0 ψ ( γ ( s, ∇ ψ ( h ) , J (1)) g = ( ∇ ψ ( h ) , D (exp y ) | exp − y ( h ) ( ˙ J (0))) g = ( ∇ ψ ( h ) , w ) g . (4.11)Similarly, ( v, D (exp − x ) | h ( w )) g = ( ∇ φ ( h ) , w ) g . Thus for any h , h ∈ B ρ ( x ) we have( ∇ φ ( h ) − ∇ ψ ( h ) − P h → h ( ∇ φ ( h ) − ∇ ψ ( h )) , w ) g = ( v, D (exp − x ) | h ( w ) − D (exp − x ) | h ( P h → h w )+ ( P y → x [ D (exp − y ) | h ( P h → h w )] − P y → x [ D (exp − y ) | h ( w )])) g . (4.12)Let (all parametrized over [0 , γ and h be the constant speed geodesics from y to x , and h to h respectively, and V and W the parallel fields along γ and h respectively with V (1) = v and W (1) = w .Then the last expression in (4.12) above can be written ˆ ˆ ∂∂q ∂∂p ( V ( p ) , [ D (exp − γ ( p ) ) | h ( q ) ] W ( q )) g dqdp = ˆ ˆ ∂∂q ( V ( p ) , ∇ ˙ γ ( p ) [ D (exp − γ ( p ) ) | h ( q ) ] W ( q )) g dqdp. Fix any local coordinates near γ ( p ) and h ( q ), then we find (below, all expressions are evaluatedat ( x, h ) = ( γ ( p ) , h ( q ))) ∇ ˙ γ ( p ) [ D (exp − γ ( p ) ) | h ( q ) ] W ( q ) = − ∇ ˙ γ ( p ) [ D h ∇ x d ( x, h ) ] W ( q )= − ∇ ˙ γ ( p ) ( g jk ( x ) ∂ x k h i d ( x, h ) W i ( q ) ∂ x j )= −
12 ˙ γ l ( p )[ ∂ x l ( g jk ( x ) ∂ x k h i d ( x, h )) + ( g rk ( x ) ∂ x k h i d ( x, h ))Γ jlr ( x )] W i ( q ) ∂ x j where here, Γ ijk are the Christoffel symbols. In particular ∇ ˙ γ ( p ) [ D (exp − γ ( p ) ) | h ( q ) ] W ( q ) is linear in W ( q ), hence we can continue calculating as ˆ ˆ ∂∂q ( V ( p ) , ∇ ˙ γ ( p ) [ D (exp − γ ( p ) ) | h ( q ) ] W ( q )) g dqdp = ˆ ˆ | ˙ γ ( p ) | ∂∂q (( ∇ ˙ γ ( p ) | ˙ γ ( p ) | [ D (exp − γ ( p ) ) | h ( q ) ]) t V ( p ) , W ( q )) g dqdp = d ( x , y ) ˆ ˆ (cid:12)(cid:12)(cid:12) ˙ h ( q ) (cid:12)(cid:12)(cid:12) ( ∇ ˙ h ( q ) | ˙ h ( q ) | ( ∇ ˙ γ ( p ) | ˙ γ ( p ) | [ D (exp − γ ( p ) ) | h ( q ) ]) t V ( p ) , W ( q )) g dqdp ≤ sup p,q k ( ∇ ˙ h ( q ) | ˙ h ( q ) | ( ∇ ˙ γ ( p ) | ˙ γ ( p ) | [ D (exp − γ ( p ) ) | h ( q ) ]) t k d ( h , h ) d ≤ Cd ( h , h ) d for some constant C > ∂ Ω and k·k is the operator norm above (againcalculating in local coordinates shows ( ∇ ˙ γ ( p ) | ˙ γ ( p ) | [ D (exp − γ ( p ) ) | h ( q ) ]) t is a linear operator). Thus recalling(4.12) we have [ ∇ ( φ − ψ )] C , ( B ρ ( x )) ≤ Cd . (4.13)Then for any h ∈ B ρ ( x ) (recall ρ from (4.9)) |∇ ( φ − ψ )( h ) | g ≤ |∇ ( φ − ψ )( x ) | g + Cd ( x , h ) d ≤ | v − ∇ ψ ( x ) | g + Cρd = | v − ∇ ψ ( x ) | g + Cd α / (1+ β )0 . (4.14)By (4.11) again we calculate for an arbitrary unit length w ∈ T x ( ∂ Ω), | ( v − ∇ ψ ( x ) , w ) g | = (cid:12)(cid:12) ( v, w − P y → x [ D (exp − y ) | x ( w )]) g (cid:12)(cid:12) ≤ | v | g (cid:12)(cid:12) w − P y → x [ D (exp − y ) | x ( w )] (cid:12)(cid:12) g ≤ Cd , for some universal C >
0. In particular, this gives | v − ∇ ψ ( x ) | g ≤ Cd , which combining with(4.14) yields k∇ ( φ − ψ ) k C ( B ρ ( x )) ≤ Cd . (4.15)Thus combining the above with (4.14) we have (cid:12)(cid:12)(cid:12) ∇ [(1 − ˜ φ )( φ − ψ )]( h ) (cid:12)(cid:12)(cid:12) g ≤ (cid:12)(cid:12)(cid:12) ∇ (1 − ˜ φ ) (cid:12)(cid:12)(cid:12) g | φ ( h ) − ψ ( h ) | g + |∇ ( φ − ψ )( h ) | g ≤ C ( d ρ + d + d α β ) ) ≤ Cd − α β ) . Finally, h ∇ ((1 − ˜ φ )( φ − ψ )) i C β ≤ k − ˜ φ k L ∞ [ ∇ ( φ − ψ )] C β + k φ − ψ k L ∞ h ∇ (1 − ˜ φ ) i C β + k∇ (1 − ˜ φ ) k L ∞ [ φ − ψ ] C β + k∇ ( φ − ψ ) k L ∞ h − ˜ φ i C β ≤ C ( ρd ρ β + d ρ β + d ρ β ) ≤ C ( d ρ β ) = Cd − α where here all of the norms are taken over B ρ ( x ). Thus we have shown that k (1 − ˜ φ )( φ − ψ ) k C ,β ( B ρ ( x )) ≤ Cd − α . Now choose α , β , and α ∈ (0 ,
1] so that α := min { α , β − α , − α } >
0, combining the finalestimate above with (4.5), (4.6), (4.7), (4.10) yields I ≤ Cd α . Finally recalling (4.3), (4.4), we will have for some universal
C > α ∈ (0 ,
1) the estimate | ( b ( x ) − P y → x b ( y ) , v ) g | ≤ Cd ( x , y ) α which in turn proves that b is locally H¨older continuous. (cid:3) The proof of Theorem 1.4.
Here we provide the proof of the control of the H¨older conti-nuity of the L´evy measure with respect to the TV norm.
Proof of Theorem 1.4.
Fix δ >
0, some x ∈ ∂ Ω, and r = δ . We assume that 2 δ < min { , inj( ∂ Ω) } where inj( ∂ Ω) is the injectivity radius of ∂ Ω. First we claim there exists α ∈ (0 ,
1) and
C > φ ≡ B r ( x ) ∩ ∂ Ω, then kI ( φ, · ) k C α ( B r ( x )) ≤ Cr k φ k L ∞ ( ∂ Ω) . (4.16)Indeed, the claim immediately follows by the comparison principle combined with [20, Corollary8.36] in the divergence form case (1.2), and in the non-divergence form case (1.3), it follows from[20, Theorem 9.31 and eq (9.71)]Now by (1.8) and density of C ,α ( ∂ Ω) in L ∞ ( ∂ Ω), it is sufficient to prove that for any φ ∈ C ,α ( ∂ Ω) with k φ k L ∞ ( ∂ Ω) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ ∂ Ω φ ( y ) χ ∂ Ω \ B δ ( x ) ( y ) µ ( x , dy ) − ˆ ∂ Ω φ ( y ) χ ∂ Ω \ B δ ( x ) ( y ) µ ( x , dy ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cd ( x , x ) α (4.17)for some C > α , whenever x , x ∈ B r ( x ).Let η k,x ∈ C ( ∂ Ω) be such that 0 ≤ η k,x ≤ ∂ Ω, with η k,x ≡ B δ ( x ) and η k,x ≡ ∂ Ω \ B δ +1 /k ( x ), and an analogous choice for η k,x . Then we find |I ( η k,x φ, x ) − I ( η k,x φ, x ) | ≤ |I ( η k,x φ, x ) − I ( η k,x φ, x ) | + |I ( φ ( η k,x − η k,x ) , x ) |≤ Cr k φ k L ∞ ( ∂ Ω) d ( x , x ) α + |I ( φ ( η k,x − η k,x ) , x ) |≤ Cr d ( x , x ) α + |I ( φ ( η k,x − η k,x ) , x ) | (4.18)where to obtain the second line we have used (4.16) and that x , x ∈ B r ( x ), along with thechoice of r ; note that by the triangle inequality we have η k,x ≡ B r ( x ).To estimate the second term in (4.18), first we note by definition, η k,x − η k,x = 0 − B δ − d ( x ,x ) ( x ). Likewise, we have η k,x − η k,x = 1 − B δ + d ( x ,x )+1 /k ( x ). Thenby Theorem 1.1 (ii), we obtain |I ( φ ( η k,x − η k,x ) , x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B δ + d ( x ,x /k ( x ) \ B δ − d ( x ,x ( x )) φ ( h )( η k,x ( h ) − η k,x ( h )) µ ( x , dh ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ B δ + d ( x ,x /k ( x ) \ B δ − d ( x ,x ( x )) d ( x , h ) − n − σ ( dh ) . Now we can consider normal coordinates centered at x , then writing s for the radial coordinateand ω for coordinates on the unit sphere S n − we can write σ = λ ( s, ω ) ds ∧ vol S n − for some realvalued function λ where vol S n − is the canonical volume form on S n − . Since ∂ Ω is compact, thereis a (possibly negative) lower bound K on the Ricci curvature, thus using standard volume formcomparison (see [42, Lemma 7.1.2]) we can calculate that λ ( s, ω ) ≤ sn n − K ( s ) ≤ s n − + n − (cid:18) max [0 , inj( ∂ Ω)] | ¨sn K | (cid:19) s n = s n − + Cs n . Here sn K ( s ) = sin ( s √ K ) √ K , K > ,s, K = 0 , sinh ( s √− K ) √− K , K < , and thus C > K , n , and the injectivity radius inj( ∂ Ω) of ∂ Ω. Then we compute ˆ B δ + d ( x ,x /k ( x ) \ B δ − d ( x ,x ( x )) d ( x , h ) − n − σ ( dh )= ˆ δ + d ( x ,x )+1 /kδ − d ( x ,x ) (cid:18) ˆ S n − s − n − λ ( s, ω ) vol S n − ( dω ) (cid:19) ds ≤ ˆ S n − vol S n − ( dω ) ˆ δ + d ( x ,x )+1 /kδ − d ( x ,x ) ( s − + Cs − ) ds ≤ C ˆ δ + d ( x ,x )+1 /kδ − d ( x ,x ) s − ds = C δ − d ( x , x ) − δ + d ( x , x ) + k ! possibly taking δ smaller. Combining this with (4.18), then taking k → ∞ and using dominatedconvergence yields (cid:12)(cid:12)(cid:12)(cid:12) ˆ ∂ Ω φ ( y ) χ ∂ Ω \ B δ ( x ) ( y ) µ ( x , dy ) − ˆ ∂ Ω φ ( y ) χ ∂ Ω \ B δ ( x ) ( y ) µ ( x , dy ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) δ d ( x , x ) α + 1 δ − d ( x , x ) − δ + d ( x , x ) (cid:19) . Finally, 1 δ − d ( x , x ) − δ + d ( x , x ) = 2 d ( x , x ) δ − d ( x , x ) ≤ d ( x , x )3 δ since d ( x , x ) ≤ r = δ/
2, hence we obtain (4.17), finishing the proof. (cid:3) Fully nonlinear equations– Proof of Theorem 1.5
In this section we treat fully nonlinear equations for (1.1) and (1.4), and we provide the proofof Theorem 1.5. We will collect some notation from Section 1.1. Recall, I is defined in (1.1) and(1.5) under the nonlinear F in (1.4). Furthermore, Theorem 1.5 will show that for φ ∈ C ,α ( ∂ Ω), I ( φ, x ) = min i max j (cid:8) f ij ( x ) + L ij ( φ, x ) (cid:9) , -to-N and Integro-Differential Operators 21 where f ij ∈ C ( ∂ Ω) and L ij are the linear operators defined as L ij ( φ, x ) = c ij ( x ) φ ( x ) + (cid:0) b ij ( x ) , ∇ φ ( x ) (cid:1) + ˆ ∂ Ω (cid:16) φ ( h ) − φ ( x ) − B r ( x ) ( h )( ∇ φ ( x ) , exp − x ( h )) g (cid:17) µ ij ( x, dh ) . (5.1)5.1. Proof of Theorem 1.5, equation (1.9).
Thanks to the Lipschitz nature of I : C ,α ( ∂ Ω) → C α ( ∂ Ω) that was established in Lemma 3.4, the min-max formula promised in Theorem 1.5 is aconsequence of [23, Theorem 1.6 and Prop 1.7] (see also [23, Theorem 1.8] which even establishesthat L ij are linear operators mapping C ,α ( ∂ Ω) → C α ( ∂ Ω)). Now we focus on the more specificbehavior of µ ij and b ij .5.2. Reduction to the extremal operators.
A very useful tool for obtaining the estimates(i-a) and (i-b) in Theorem 1.5 is the reduction from a general F in (1.4) to the particular instanceof the Pucci operator, F = M − . This is a consequence of the representation of the extremaloperators of I in terms of the D-to-N for M − , which appeared in Lemma 3.3. Specifically, werecord the result of [23, Prop 4.35] as it pertains to I in this work. As the proof of this propositionis not particular to the D-to-N mapping, we refer to [23, Sec 4.6] for its proof. Proposition 5.1 (see Proposition 4.35, Sec 4.6 of [23]) . If L ij is any one of the collection oflinear operators appearing in Theorem 1.5, defined in (5.1), then for all φ ∈ C c ( ∂ Ω) , the followingestimate holds: M − ( φ, x ) ≤ L ij ( φ, x ) ≤ M + ( φ, x ) . Here, M ± are the extremal operators defined in Lemma 3.3. Proposition 5.1 means that in order to establish the estimates in Theorem 1.5, we can focus onobtaining, e.g. lower bounds for M ± ( φ, x ). This is a welcome simplification to the problem, forexample because M ± (for equation (1.1)) are convex/concave as well as rotation and translationinvariant, and they enjoy good regularity theory ( C ,α boundary data produces C ,α ′ solutions).5.3. The ring estimate, Theorem 1.5 (i-a).
Here we provide the proof of the ring estimatethat appears in Theorem 1.5 (i-a).
Proof of Theorem 1.5 part (i-a).
First, we note that x ∈ ∂ Ω is just a parameter, and a translationof the equation (1.1) so that x = 0 does not change any of the assumptions on F . Thus, withoutloss of generality, we take x = 0 ∈ ∂ Ω. We will obtain the desired ring estimate by rescalingthe domain in (1.1) from Ω to a larger set, (1 /r )Ω, and representing U φ in Ω as a rescaling of anappropriate function, ˜ U ˜ φ , in (1 /r )Ω. The advantage here is to utilize the fact that ∂ ((1 /r )Ω) isbecoming flat in a C fashion under this scaling, and so we can use solutions in one fixed domainto build appropriate sub and super solutions for equations in (1 /r )Ω. We now proceed with theconstruction.Thanks to Lemma 2.5, we will work with functions and sets in R n +1 and actually show a relatedestimate (which is no harm when r is small). When B n +1 r ⊂ R n +1 is the usual ball in R n +1 , wewill prove: C r − ≤ µ ij ( x, ( B n +1(7 / r \ B n +1(5 / r ) ∩ ∂ Ω) (5.2)and µ ij ( x, ( B n +1(9 / r \ B n +1(3 / r ) ∩ ∂ Ω) ≤ C r − . (5.3) Thus, for ease of presentation let us introduce the notation for respectively the small and big rings: R Sr := ( B n +1(7 / r \ B n +1(5 / r ) ∩ ∂ Ω and R Br := ( B n +1(9 / r \ B n +1(3 / r ) ∩ ∂ Ω . The reason for this simplification is to be able to work with φ that are actually defined in all of R n +1 , and use their restrictions to various submanifolds as Dirichlet data. To this end, let φ rl and φ ru be C ( R n +1 ) lower and upper barrier functions such that0 ≤ φ rl ≤ R Sr ≤ R Br ≤ φ ru , (5.4)and furthermore, just for concreteness, we assume φ lr , φ ur satisfy ( φ rl ≡ B n +1(13 / r \ B n +1(11 / r φ rl ≡ B n +1(14 / r \ B n +1(10 / r (5.5)and ( φ ru ≡ R Br φ ru ≡ B n +1(19 / r \ B n +1(5 / r . (5.6)Thus we see that ˆ ∂ Ω \{ x } φ rl ( y ) µ ij ( x, dy ) ≤ ˆ ∂ Ω \{ x } R Sr ( y ) µ ij ( x, dy ) (5.7)and ≤ ˆ ∂ Ω \{ x } R Br ( y ) µ ij ( x, dy ) ≤ ˆ ∂ Ω \{ x } φ ru ( y ) µ ij ( x, dy ) . (5.8)Furthermore, since φ rl ( x ) = φ ru ( x ) = 0 and ∇ φ rl ( x ) = ∇ φ ru ( x ) = 0 , we see that the operators in (5.1) simplify to L ij ( φ rl , x ) = ˆ ∂ Ω φ rl ( y ) µ ij ( x, dy ) and L ij ( φ ru , x ) = ˆ ∂ Ω φ ru ( y ) µ ij ( x, dy ) . (5.9)Thus, to conclude (5.2) and (5.3), it suffices, via Proposition 5.1 combined with (5.9), (5.7), and(5.8) to show the same bounds for the normal derivatives of the functions U φ rl and U φ ru that solve(1.1) with respectively F = M − and F = M + .Now, we record our target to achieve (5.2) and (5.3). Assume that U φ lr and U φ ur are respectivelythe solutions of (1.1) for F = M − and F = M + with Dirichlet data given respectively by φ rl | ∂ Ω and φ ru | ∂ Ω . We will show goal: there are universal constants so that C r − ≤ ∂ ν U φ rl (0) and ∂ ν U φ ru (0) ≤ C r − . (5.10)We will give the details for the bound on ∂ ν U φ rl , and the upper bound for ∂ ν U φ ru will followanalogously.We will represent U φ rl as a rescaling of a particular function, ˜ U , in a larger domain, by defining˜Ω r = (1 /r )Ω , M − ( ˜ U ) = 0 in ˜Ω r and ˜ U | ∂ ˜Ω r = φ l | ∂ ˜Ω r , and U φ lr ( y ) = ˜ U ( yr ) for y ∈ Ω . -to-N and Integro-Differential Operators 23 This means that M − ( φ l | ∂ Ω , y ) = ∂ ν U φ lr ( y ) = r − ∂ ν ˜ U ( y ) . Thus, our new goal will be to show that unscaled goal: C ≤ ∂ ν ˜ U (0) . (5.11)In order to get a lower estimate on ∂ ν ˜ U that is truly independent of r , we will use an auxiliaryfunction that is independent of r and defined in a fixed domain, independent of r . Let us call the“half” ball, B +10 (0) := { y ∈ R n +1 : y · ν (0) > } ∩ B n +110 . Then we can define the function ˜ V as the unique solution of M − ( ˜ V ) = 0 in B +10 , and ˜ V | ∂B +10 = φ l | ∂B +10 . The advantage of ˜ V is that it is independent of r , and so as long as we can show that ˜ U − ˜ V issmall enough in the C ,α sense, then we will be able to conclude the auxiliary goal in (5.11).In order to get the estimate between ˜ U and ˜ V , we must introduce two more auxiliary functions.The first is ˜ W r , defined in the domain ˜ B , ˜ B := ˜Ω r ∩ B +10 , and M − ( ˜ W r ) = 0 in ˜ B, and ˜ W r | ∂ ˜ B = φ l | ∂ ˜ B . Thus, since ˜
U > r , we see that ˜ W r is a subsolution (including ordering of boundarydata) to the equation for ˜ U (or vice-versa, ˜ U is a supersolution for the equation for ˜ W r ), henceby the comparison principle, ˜ U ≥ ˜ W r in ˜ B, and ∂ ν ˜ U (0) ≥ ∂ ν ˜ W r (0) . Now, to conclude, we will show a lower bound for ∂ ν ˜ W r (0).We note that the distance between the half space determined by the tangent to ∂ ˜Ω r in B n +110 , { y ∈ R n +1 : y · ν (0) > } ∩ B n +110 , and to ∂ ˜Ω r ∩ B n +110 is vanishing as r → Cr ). Furthermore, by the boundary estimates in [48, Theorem 1.1], we know that k ˜ U k C ,α (˜Ω r ∩ B n +110 ) ≤ C k φ l | ∂ ˜Ω r k C ,α ≤ C, (note, by the Evans-Krylov Theorem, ˜ U is actually C ,γ , but we only invoke estimates for ˜ U and ∇ ˜ U ). Hence, by the flattening of ∂ ˜Ω r as r →
0, we see that on the “lower” boundary of B +10 , wecan make ˜ U and φ l close: k ˜ U − φ l k C ,α ( { y ∈ R n +1 : y · ν (0)=0 }∩ B n +110 ) → r → . This means that if we define the function ˜ Z = ˜ W r − ˜ V , then in the common domain of their equations, we have, by the properties of viscosity solutions M − ( ˜ Z ) ≤ M + ( ˜ Z ) ≥ r ∩ B +10 , and thanks to the boundary values for ˜ V k ˜ Z | ∂ (˜Ω r ∩ B +10 ) k C ,α → r → . Furthermore, since ˜ V is independent of r , and since ˜ V attains a minimum at y = 0 ∈ ∂B +10 by theHopf principle, we know that for a C that depends only on universal parameters and the choiceof φ l , ∂ ν ˜ V (0) = C > . Hence, taking R small enough (recall R from Theorem 1.5 (i-a)), so that for r ≤ R , we have k ˜ Z | ∂ (˜Ω r ∩ B +10 ) k C ,α < C , we can then conclude for these r ≤ R that ∂ ν ˜ W r (0) ≥ C , and also, by the above comparison of ˜ W r and ˜ U , ∂ ν ˜ U (0) ≥ C C , where C is a universal constant. As noted earlier, rescaling ˜ U , gives the lower bound.The proof of the upper bound follows analogously. Instead of using M − to define the functions˜ U , ˜ W r , ˜ V , ˜ Z , we will use the operator M + . Also, at the stage of using comparison to switchfrom ˜ U to ˜ W r , it will be useful to use boundary data that is identically 1 outside of B n +13 / so that˜ W r can serve as a supersolution for ˜ U . Thus, this same function will be used to determine theboundary values of ˜ V , instead of φ u , which would have been the direct analog of the argument.Everything else follows similarly. (cid:3) A lower bound for µ ab ( x, · ) in Theorem 1.5 part (i)(b). Next, we prove the lower boundfor µ ab in Theorem 1.5 (i-b). Our approach will be to work in the context of linear equations withsmooth coefficients, and invoke some techniques and results about the related Green’s functionsfrom e.g. [36]. In order to transfer results between fully nonlinear equations and equations withsmooth coefficients, we will collect various facts and observations from the literature. This firstfact is a technique for approximating solutions of fully nonlinear equations by those of linearequations with smooth coefficients. It is more or less well known to specialists, but there does notseem to be any standard reference. Here we present the technique as used by Feldman [19, Proofof Prop. 2.2], where it is proved in complete detail. Since this is nearly exactly as implementedin [19], we simply list a sketch of the steps without detailed justification/explanation. Lemma 5.2 (Smooth Linear Approximation) . Any solution of Pucci’s equation can be approxi-mated by solutions of linear equations with smooth coefficients and the same ellipticity bounds.Given φ ∈ C ( ∂ Ω) and U φ solving (1.1) and (1.4) with F ( D U, x ) = M − ( D U ) , there existsa family of coefficients, A δ ( x ) , depending on U φ , which are uniformly elliptic all with the sameconstants ( λ, Λ) and smooth in x , such that for U δφ solving ( tr( A δ ( x ) D U δφ ) = 0 , in Ω U δφ = φ on ∂ Ω , we recover k U δφ − U φ k L ∞ (Ω) → as δ → . Proof of Lemma 5.2.
Again, as mentioned above, we present only a sketch of the proof that comesfrom [19, Prop. 2.2].Here are the steps: -to-N and Integro-Differential Operators 25 (1) Approximate M − by smooth concave functions, M − ,k , giving w k that solve the smoothedequation. For eventual limiting operations via the stability of viscosity solutions, thisrequires that M − ,k → M − uniformly on compact subsets of S (( n + 1) × ( n + 1))(2) Linearize M − ,k over w k , and use the fact that w k are C ,α (Ω) (see e.g. [9]), which can bedone explicitly as a ki,j ( x ) := ˆ ∂ M − ,k ∂P i,j ( sD w k ( x )) ds. (3) Extend a ki,j to all of R d +1 as simply a ki,j ( x ) = δ i,j for all x Ω (note δ is the Kroneckerdelta symbol).(4) Mollify a ki,j to be smooth, denoting them as a k,mi,j .(5) Taking the matrix A k,m = ( a k,mi,j ), solve the equation ( tr( A k,m D w k,m ) = 0 in Ω w k,m = φ on ∂ Ω . (6) Confirm that there exists a subsequence w k → U φ uniformly in Ω as k → ∞ , as well as asubsequence w k,m → w k uniformly in Ω for k fixed and m → ∞ . In both cases, one caninvoke, for example C α estimates, as all of these functions are uniformly bounded with acommon bound. The first convergence and stability result uses regular viscosity solutionstheory, and the second convergence uses the the L p viscosity solutions theory in e.g. [6].We note that the limit in both cases uses the fact that viscosity solutions are stable andthat the limit equations have unique solutions.We briefly remark that the reason for invoking the L p theory is that it is not known how goodare the coefficients a ki,j in the vicinity of ∂ Ω. It seems reasonable in this lemma to want to keepthe same boundary values throughout the whole process. We note that if it so happens that φ ∈ C ,α ( ∂ Ω), then one can use regular viscosity solutions for both convergence arguments, asthis would produce w k ∈ C ,α (Ω), and hence a ki,j ∈ C α (Ω). (cid:3) This next lemma is a simple exercise for constructing a sequence of balls linking points in Ω,each of whose radius is a (fixed) multiple of the previous. For C domains, it is a simpler propertythan the Harnack chains that are used in [31], but we keep the same name nonetheless. We omitthe proof. Lemma 5.3 (The Harnack chain distance) . If ∂ Ω is bounded and C , then there is a universal R so that if r > is fixed, and y ∈ Ω and x ∈ Ω ∩ B R ( y ) , with d ( y, ∂ Ω) > r and d ( x, ∂ Ω) > r then x and y can be linked by a Harnack chain based on balls of multiples of radius, r , so that N = { balls in the chain } ≤ C log( C | x − y | r ) . Here, the constants C and C are independentfrom r , and they depend only on n , λ , Λ .(We note that by Harnack chain based on balls of radius r , we mean a sequence of balls thatsuccessively overlap, twice each is contained in Ω , the first contains y and the last contains x , andall of their radii are multiples of r .) In the next couple of results, in investigating the Harmonic measure of B n +1 r ( h ) ∩ Ω, it will beuseful to use an auxiliary ball that is actually inside Ω, and has both a size and distance to ∂ Ωthat are comparable to r . We call this ball, ˜ B r , and we record its definition here: Definition 5.4.
Given h ∈ ∂ Ω and B n +1 r ( h ) ∩ Ω , the auxiliary ball is ˜ B r = B n +1 r ( h + 4 r · ν ( h )) (recall ν ( h ) is the inward normal vector at h ). We will call ˆ h = h + 4 r · ν ( h ) The next two results apply to any operator of the form L A u ( x ) = tr( A ( x ) D u ( x )) such that A is smooth and uniformly λ, Λ-elliptic. The resulting bounds depend only on dimension and λ, Λ.However, we only use them for the A δ produced by Lemma 5.2, and so that is how we will presenttheir results. The next two lemmas are a blending of ideas from [36, Section 5] and [10, AppendixB]. Lemma 5.5.
Let A δ be as in Lemma 5.2 and G δ be the Green’s function for A δ in Ω . If x ∈ Ω , h ∈ ∂ Ω , | x − h | = l , r is small enough, d ( x, ∂ Ω) > r , and ˜ B r is the ball B n +1 r ( h + 4 rν ( h )) ⊂ Ω ,then there exists a universal η ≥ n and C so that C ( rl ) η ≤ r ˆ ˜ B r G δ ( x, z ) dz. Lemma 5.6.
Let A δ be as in Lemma 5.2, G δ the Green’s function for A δ in Ω , and ω δ be theharmonic measure for A δ in Ω . If x B n +1 r ( h + 4 ν ( h )) , and r is small enough, ω x ( B n +1 r ( h ) ∩ ∂ Ω) ≥ r ˆ ˜ B r G δ ( x, z ) dy, where as above we are using ˜ B r = B n +1 r ( h + 4 ν ( h )) . Remark 5.7.
We believe it may be worth noting that although the result claimed in Lemma 5.5,especially if x approaches ∂ Ω , seems strange, there is no contradiction in the inequality. Eventhough one expects ´ ˜ B r G δ ( x, z ) dz ≤ cr d ( x, ∂ Ω) (as will be apparent from the subsequent proofs,combined with boundary behavior), there is a restriction for the Harnack chain that d ( x, ∂ Ω) ≥ r .Thus, in the worst case, if we take d ( x, ∂ Ω) = 2 r , we see that Lemma 5.5 will imply c ( r/l ) η ≤ d ( x, ∂ Ω) = 2 r , and this inequality does not cause a problem. The usefulness of the inequality willbe when d ( x, ∂ Ω) is of order l , which is much larger than r . First, we will prove Lemma 5.5.
Proof of Lemma 5.5.
Let x , h , and r be fixed as in the statement of the lemma. Let us define thefunction w ( y ) = ˆ ˜ B r G δ ( y, z ) dz = ˆ Ω ˜ B r ( z ) G δ ( y, z ) dz. (5.12)That is to say, that by definition, w is the unique function that solves ( L A δ w = − ˜ B r in Ω w = 0 on ∂ Ω . (5.13)First, using a comparison argument, we will get a lower bound on w in ˜ B r . Then we will iterateit using a Harnack chain until we reach x .Let us recall ˆ h = h + 4 r · ν ( h ). We note that for an appropriate choice of θ (depending only on n , λ , Λ), the function p ( y ) = θ ( r − (cid:12)(cid:12)(cid:12) y − ˆ h (cid:12)(cid:12)(cid:12) ) satisfies L A δ p ≥ − ˜ B r in ˜ B r and p = 0 on ∂ ˜ B r . Thus, by comparison, we have obtained that w (ˆ h ) ≥ θr , in 12 ˜ B r = B n +1 r/ (ˆ h ) . -to-N and Integro-Differential Operators 27 Now, using a barrier for L a δ u = 0 in B n +12 r (ˆ h ) \ B n +1 r/ (ˆ h ), we can conclude that w ≥ c θr in B n +13 r/ ˆ h. By iterating Harnack’s inequality in a Harnack chain of balls proportional to ˜ B r , we see that if N is the number of such balls required to link ˆ h to x , then there is a universal C > w ( x ) ≥ (cid:18) C (cid:19) N θr . Thus, invoking the Harnack chain bound in Lemma 5.3, we see that w ( x ) ≥ (cid:18) C (cid:19) C log( C lr ) θr . By setting η as, η = − log (cid:18) C (cid:19) C ! , we see then that (cid:18) C (cid:19) C log( C lr ) = (cid:18) rC l (cid:19) η . Hence, we see that for another, universal, ˜ C , w ( x ) ≥ ˜ C (cid:16) rl (cid:17) η r . Dividing by r , relabeling ˜ C , and recalling (5.12) conclude the lemma. (cid:3) Before we give a proof of Lemma 5.6, we will need a result about a barrier function. We willuse the fact that because Ω has the uniform exterior ball condition, given a point, h ∈ ∂ Ω, we canchoose an annulus, for constants c and R that depend only on Ω, so that for an appropriate y ,Ω ⊂ B n +1 R ( y ) \ B n +1 c ( y ) and B n +1 c ( y ) is tangent to ∂ Ω at h ∈ ∂ Ω. Lemma 5.8.
Assume that c > and r > are given, with r < c . There exists a function, ψ ,that solves in the viscosity sense, M + ( ψ ) ≤ − for c − r < | x | < c + 5 r, M + ( ψ ) ≤ for c − r < | x | , with ψ ≥ in R n +1 and sup R n +1 ( ψ ) ≤ cr . The proof of Lemma 5.8 is an explicit calculation, and we defer its proof until after the proofof Lemma 5.6.
Proof of Lemma 5.6.
This proof appears, for example, in the proof of [36, Lemma 5.18]. We giveslightly more detail here. We recall that˜ B r = B n +1 r ( h + 4 ν ( h )) . Let x , h , and r be fixed as in the statement of the lemma. Let us recall the function, w , asdescribed in (5.12) and (5.13). For simplicity, for y ∈ Ω let us just call v ( y ) = ω y ( B n +1 r ( h ) ∩ ∂ Ω). We know from the definitionof harmonic measure that v satisfies ( L A δ v = 0 in Ω v = B n +1 r ( h ) ∩ Ω on ∂ Ω . We wish to establish the lemma by the following two claims, followed by the maximum principlein Ω \ ˜ B r , as v and w satisfy the ordering v | ∂ Ω ≥ w | ∂ Ω . Claim 1: for some universal c > w satisfies the estimate sup ˜ B r w ≤ cr .Claim 2: for some universal c >
0, inf ˜ B r v ≥ c .First, we address claim 1. We use the exterior ball condition for Ω with balls of radius, c .Thus, there is some ˜ h Ω so that B n +1 c (˜ h ) ⊂ Ω C and is tangent to ∂ Ω at h . After an appropriatetranslation, we see that the function, ψ , from Lemma 5.8 can be made to be a super solution inthe set, | y | > c − r (as M + ψ ≥ L A δ ψ , by definition of M + ), which contains Ω. Furthermore, byconstruction, after a translation, we will have M + ψ ≤ − B r . Hence, this translation of ψ is asuper solution for the same equation as w , and that by construction, ψ ≥ ∂ Ω. Hence claim1 follows from the comparison theorem for L A δ in Ω and the estimate that sup( ψ ) ≤ cr .To see why claim 2 is true, we invoke [36, Lemma 5.3] (which also comes from [18]), for thefunction (1 − v ), first in B r/ ( h ) ∩ Ω. That is to say that since sup B r ∩ Ω (1 − v ) ≤
1, we see thatfor some universal ¯ α for x = h + ( r ν ) , (1 − v ( x )) ≤ (cid:18) (cid:19) ¯ α . In other words, v ( x ) ≥ − (1 / ¯ α . Now, for example, with x = h + 58 ν and x = h + 78 ν, using a ball of radius 3 r/
16, we see that Harnack’s inequality applies so that1 − ( 12 ) ¯ α ≤ v ( x ) ≤ sup B r/ ( x ) v ≤ C inf B r/ ( x ) ≤ Cv ( x )Repeating this process three more times (with slightly larger radii) gives that if y ∈ ˜ B r , inf ˜ B r v ≥ C .Now, to conclude the theorem, we see that after multiplying by an appropriate universal con-stant, v ( y ) ≥ Cr w ( y ) for all y ∈ ˜ B r . Hence, by the above observation that v and w solve the same equation (with zero right hand side)in Ω \ ˜ B r , we conclude that v ( y ) ≥ Cr w ( y ) for all y ∈ ˜Ω , and this implies them lemma, taking y = x . (cid:3) Finally, we are in a position put the steps together to prove Theorem 1.5 part (i)(b). -to-N and Integro-Differential Operators 29
Proof of Theorem 1.5 part (i)(b).
We first assume that x ∈ ∂ Ω, h ∈ ∂ Ω, r > x = h , and r < ( d ( x, h )) /
10. For this part of the proof, it is easiest to assume that d ( x, h ) issmall enough so that if | x − h | = l , then x + lν ( x ) ∈ Ω and B n +12 r ( x + lν ( x )) ⊂ Ω. This is nota restriction, as we have already assumed that Ω is bounded and ∂ Ω is C . (We also note anintentional switch to using | x − h | in this section as we can assume this is comparable to d ( x, h ).)We note that just as above, we shall assume that φ is smooth and φ ≥ B n +1 r ( h ) ∩ Ω . The result will follow by taking a sequence of such φ , decreasing to B r ( h ) , but we suppress thesequence for now to keep the notation to a minimum. The key properties of φ that we assume are φ ( x ) = 0 , and ∇ φ ( x ) = 0 . We now remind the reader that for this part of the theorem, if µ ij are as in (5.1) (which is givenby the first part of the theorem), we must show that µ ab ( B n +1 r ( h ) ∩ Ω) ≥ cr η d ( x, h ) η +1 . According to our choice of φ , combined with the formula in (5.1), and that ∇ φ ( x ) = φ ( x ) = 0, L ij ( φ, x ) = ˆ ∂ Ω φ ( z ) µ ij ( x, dz ) . Thus, in other words, our goal can be recast as showing L ij ( φ, x ) ≥ cr η d ( x, h ) η +1 , and hence since the lower bound uses only that φ ≥ B n +1 r ( r ) ∩ Ω , the claim will follow by letting φ decrease pointwise to B n +1 r ( r ) ∩ Ω . Again, as above, this lower bound can be obtained by findinga lower bound for the extremal operators, per Proposition 5.1. Thus, Proposition 5.1 shows thefollowing estimate will suffice: M − ( φ, x ) ≥ cr η d ( x, h ) η +1 , (5.14)where we recall the D-to-N extremal operator, M − , defined in 3.1.Now, assume that U φ is the unique solution of ( M − ( U φ , y ) = 0 in Ω U φ = φ on ∂ Ω . (5.15)We will focus on the values of U φ (ˆ x ), where ˆ x is chosen so thatˆ x = x + lν ( x ) recall ( l = | x − h | ) . Invoking barriers, such as those of the form C ( d ( y, ∂ Ω) + cd ( y, ∂ Ω) ), which are subsolutions to(5.15), we see that if we can show that U φ (ˆ x ) ≥ C (cid:16) rl (cid:17) η , (5.16)then it follows, with linearly growing barriers, that U φ ( y ) ≥ C d ( y, ∂ Ω) (cid:0) rl (cid:1) η l . Hence, as soon as we obtain (5.16), it follows that ∂ ν U φ ( x ) ≥ Cr η l η +1 , which is exactly what is needed, via M − ( φ, x ) to obtain (5.14).Given that the goal in (5.14) is a pointwise bound, and given that we can approximate U φ and(5.15) via solutions to linear equations to smooth coefficients using Lemma 5.2, it suffices to showthat the U δφ in Lemma 5.2 also enjoys U δφ (ˆ x ) ≥ C (cid:16) rl (cid:17) η . However, this last equation follows immediately from Lemmas 5.5 and 5.6, combined with thefact that φ ≥ B n +1 r ( h ) ∩ Ω and using the comparison principle for the functions U δφ ( y ) and v ( y ) = ω y ( B n +1 r ( h ) ∩ ∂ Ω). (cid:3)
Remark 5.9.
The reader should see, through the details of the proof, that in the nice case oflinear equations with H¨older coefficients, we will recover η = n . Indeed, in Lemma 5.5, this resultfollows immediately from the estimates on Green’s functions invoked in Section 4. However, in theabsence of these estimates, the seemingly only available tool was Harnack’s inequality, at whichpoint multiple invocations of it will lead to some η that is expected to be significantly larger than n (this is in Lemma 5.5). This means that in the nonlinear setting, the L´evy measures may assigna much smaller mass to balls than in the linear case with H¨older coefficients. To conclude this section, we will give the calculation that leads to the barrier in Lemma 5.8.
Proposition 5.10.
Given any b > , there exists ε > and a < / that are independent from b and depend only on λ , Λ , n , such that there exists a function, f , that solves in the viscositysense: f ≥ in R n +1 M + ( f ) ≤ for | x | > a b and M + ( f ) ≤ − ε for a b < | x | < b . proof of Proposition 5.1. We first begin with the function, g , defined as g ( t ) = ( − t ( t − b ) if t ∈ [0 , b ) b if t ∈ [ b , ∞ ) . We will construct f as f ( x ) = g ( | x | ) . Thus, computing derivatives, we see that ∂ f∂x i ∂x j ( x ) = g ′′ ( | x | ) x i x j | x | + g ′ ( | x | ) (cid:16) | x | − x i | x | (cid:17) if i = jg ′′ ( | x | ) x i x j | x | − g ′ ( | x | ) x i x j | x | if i = j. Furthermore, since f is a radial function and M + is a rotationally invariant operator, it sufficesto check the equation only for M + ( f, te ). First, we do this for the case of t ∈ (0 , b ). -to-N and Integro-Differential Operators 31 Plugging in x = te to the second derivatives of f shows ∂ f∂x i ∂x j ( te ) = g ′′ ( t ) if i = j = 1 g ′ ( t ) t − if i = j = 10 otherwise . Thus, computing M + ( f, te ) (recall x ∈ R n +1 ), we get M + ( f, te ) = n Λ g ′ ( t ) t − + λg ′′ ( t )= n Λ( − bt ) − λ, where we note we have used that g ′ ( t ) ≥ t ∈ (0 , b ). We now see thatlim t → ( b/ g ′ ( t ) t − = 0 , and hence lim t → ( b/ M + ( f, te ) = − λ. Thus, to be concrete, we may choose ε = λ , from which the existence of a < follows from thefact that M + ( f, te ) is strictly decreasing for t < b and sufficiently close to b .The previous calculation verifies the claimed inequality for M + ( f, x ) for | x | ∈ ( a b, b ). In orderto confirm the remaining cases of x , we simply note that at all x with | x | > a b , we have that f is either twice differentiable at x , or any test function, φ , must satisfy D φ ( x ) ≤ f − φ attains a minimum. Hence we obtain the the equation for | x | ∈ ( a b, ∞ ). (We note tothe reader that avoiding a neighborhood of x = 0 is intentional, as f can be touched from belowby functions with a positive Hessian there.) This concludes our proof. (cid:3) Now that we have the basic function, f , the proof of Lemma 5.8 follows as a simple corollary. Proof of Lemma 5.8.
Starting with the function, f , and b = 12 r , from Proposition 5.10, thefunction ψ can be constructed using suitable choices of a dilation, a shift, and a multiplicationby a constant. Furthermore, all of these operations depend upon and change f by only factorsthat are universal in the sense of depending on the exterior ball radius, c , and λ , Λ, n . Since, byconstruction, f enjoys the bound, sup f ≤ b /
4, we see that after these transformations, we willretain ψ ≤ cr for some universal c . (cid:3) Comments on more general boundary conditions
For elliptic equations, such as (1.1) with F as in (1.2)–(1.4), two of the most natural boundaryconditions (depending upon whom is asked) would be U | ∂ Ω = φ and ∂ ν U = g . This paper, of coursegives a description of the link between the two. However, the Neumann condition, ∂ ν U = g , is justthe prototype of this family, and there are many other possibilities, such as oblique, capillarity,geometric, and Robin: B ( x ) · ∇ U ( x ) = g ( x ) , with B ( x ) · ν ( x ) ≥ λ > ∂ ν U ( x ) = g ( x ) q |∇ U ( x ) | ∂ ν U ( x ) = g ( x ) |∇ U ( x ) | ∂ ν U ( x ) = g ( x ) U ( x ) . In all cases, these types of boundary conditions can be written generically as G ( x, U, ∇ U ) = 0 , where G is increasing with respect to ∂ ν U . The requirement that G is increasing comes from thefact that G is used in conjunction with an elliptic equation, for which the comparison principle isessential, and hence the relevant G all also enjoy this monotonicity property with respect to ∂ ν U .This means the standard assumption is that G ( x, r, p + cν ( x )) − G ( x, r, p ) ≥ λc (or, more generally , > , combined with natural growth restrictions jointly in the x, r, p variables. There are many workson this topic, but we point to Barles [1], Lieberman-Trudinger [39], and [40] for a sample of resultsand more references.The key point about these more general Neumann-type operators, G , is that they all obey theglobal comparison property, and under natural ellipticity assumptions, it is not hard to check thatthey too, just as with I , will be Lipschitz mappings of C ,α ( ∂ Ω) → C α ( ∂ Ω). What this means inthe context of our operator, I , is that many, if not all of the results of Theorems 1.1 – 1.5 shouldhave direct analogs to the case of the operator, G , which is defined as G ( φ, x ) = G ( x, φ ( x ) , ∇ U φ ( x )) , where U φ is as in (1.1) and G is as above.7. Open Questions
We believe there are at least a few natural open questions that arise as a result of Theorems1.1–1.5. Here we briefly explain some of them.
Lack of symmetry.
Even in the case of a flat domain, Ω = R n +1+ , one does not expect theresulting integro-differential operators to have symmetric kernels in the sense that k ( x, x − h ) = k ( x, x + h ) (or in the nonlinear case µ ( x, B r ( x + h )) = µ ( x, B r ( x − h ))). In the linear case, one cansee this immediately from the fact that if you set x = 0, then, except in special circumstances, onewill have U φ = U φ ( −· ) (or, more importantly, one would need equality with reflected data at all x ). This suggests that both a nonzero drift term, b , and the lack of symmetry of µ is inevitable.Thus, going forward, it will be important to characterize this lack of symmetry in a precise way.Furthermore, we suggest that this drift and lack of symmetry will also be present in the case ofnonlinear equations, even for the D-to-N operators for the Pucci operators. Does it correspond toany assumptions that are similar to those in [11], [13], or [45]? Or is it a different type altogether?Furthermore, there will be another source that destroys the symmetry of the L´evy measuresfrom the curvature of ∂ Ω when ∂ Ω is not flat. This effect also needs to be made precise.
Regularity theory on manifolds.
As mentioned in the introduction, one of the attractivefeatures of viewing some operators from an integro-differential viewpoint is the possibility to invokeregularity results that depend on minimal assumptions on the L´evy measure (these are referred toas Krylov-Safonov type results in Section 1.4). As it currently stands, to the best of our knowledge,there seem to be no such results when Ω is anything other than R n +1+ (meaning I acts on functionson R n ). It should be useful in the future to have analogs of the results mentioned in Section 1.4,appropriately modified to account for the correct assumptions that would be found by making thelack of symmetry precise. Regularity theory with different lower bounds on µ . The property (i)-b of the integro-differential operators resulting from Theorem 1.5 is new for the existing literature. Presumablyregularity results are obtainable for such situations, but at the moment, it is a completely openquestion. -to-N and Integro-Differential Operators 33
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Department of Mathematics, University of Massachusetts, Amherst, Amherst, MA 90095
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