Remarks on the Nonlocal Dirichlet Problem

We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.


Introduction
The aim of this article is to provide two results on translation-invariant integrodifferential operators, which are not surprising but have not been systematically covered in the literature. Let us briefly explain these results in case of the classical Laplace operator.
The classical result of Weyl says the following. Assume R is an open set, , and D is a Schwartz distribution satisfying in the distributional sense, i.e. for every . Then and in .
This is the starting point for our analysis. The first aim is to study distributional solutions to nonlocal boundary value problems of the form L in in where L is an integrodifferential operator generating a unimodal Lévy process. Our second aim is to provide sufficient conditions such that distributional solutions to the nonlocal Dirichlet problem are twice differentiable in the classical sense. In case of the Laplace operator, it is well known that Dini continuity of R, i.e. finiteness of the integral 1 0 d for the modulus of continuity , implies that the distributional solution to the classical Dirichlet problem satisfies 2 . On the other hand, one can construct a continuous function 1 R and a distribution D 1 such that in the distributional sense, but 2 1 . These observations have been made long time ago [26]. They have been extended to non-translation-invariant operators by several authors [12,33] and to nonlinear problems [14,31]. Note that there are many more related contributions including treatments of partial differential equations on non-smooth domains. In the present work we treat the simple linear case for a general class of nonlocal operators generating isotropic unimodal Lévy processes.
Let us introduce the objects of our study and formulate our main results. Let The function induces a measure d d , which is called the Lévy measure. Note that we use the same symbol for the measure as well as for the density. We study operators of the form L lim 0 d . (1.1) This expression is well defined if is sufficiently regular in the neighbourhood of R and satisfies some integrability condition at infinity. We recall that for 0 2 and d d with some appropriate constant , the operator L equals the fractional Laplace operator 2 on 2 R . The regularity theory of such operators has been intensively studied recently. For instance, it is well known [3,18,[36][37][38] that the solution of 2 with belongs to provided that neither nor is an integer. A similar result in more general setting is derived in [2]. We adopt a common convention and identify the Lévy measure with its radial profile, i.e. , R . Our standing assumption is that is a non-zero, nonincreasing radial function and that there exists a Lévy measure resp. a density such that for 0 , for 0 and 1 0 (1.2) for some 1 and 0 0. We remark that without loss of generality we may and do assume that 0 is such that the function 0 The condition L 1 R is the integrability condition needed to ensure well-posedness in the definition of L in distributional sense. Note that the shift of 0 results in a different constant in the integral above but does not affect its finiteness. Given an open set, we denote by resp. the usual Green resp. the Poisson operator associated with L , cf. Section 2. For a definition of the mean-value property and the Kato class K and K see Definition 2.1 and Definition 2.3 below. Here is our first result. The theorem above says that the distributional solution of Eq. 1.4 is unique up to a function with the mean-value property. If, additionally, is a Lipschitz domain and we impose some regularity, then there is a unique solution among all solutions which are bounded close to the boundary. Observe that we do not assume boundedness of , so, in general, may be unbounded outside of , even close to the boundary. Here bounded close to the boundary means that the restriction of to the set is bounded close to the boundary. Boundedness of , , would suffice, of course. We highlight that in general, there are functions with the mean-value property which are unbounded near to the boundary (see e.g. [6,Lemma 6]), thus, in order to address the uniqueness problem, we need to restrict ourselves to solutions bounded close to the boundary. Observe that in the first part of the theorem we do not say anything about existence of solutions; we are able to establish it only after additional assumptions on and . A more expanded discussion on the existence of solutions for problems driven by a non-local operator can be found in [8]. Note that, in the case where L equals the fractional Laplace operator, similar results like Theorem 1.1 are proved in [7]. A result similar to Theorem 1.1 has recently been proved in [29]. The authors consider a smaller class of operators and concentrate on viscosity solutions instead of distributional ones.
Variational solutions to nonlocal operators have been studied by several authors, e.g., in [17,40]. The problem to determine appropriate function spaces for the data leads to the notion of nonlocal traces spaces introduced in [15]. It is interesting that the study of Dirichlet problems for nonlocal operators leads to new questions regarding the theory of function spaces.
The formulation of our second main result requires some further preparation. They are rather technical because we cover a large class of translation-invariant operators. The similar condition to the following appears in [8].
is twice continuously differentiable and there is a positive constant such that for 0 .
(A) and Eq. 1.2 are essential for proving that functions with the mean-value property are twice continuously differentiable, see Lemma 2.2. We emphasize that in general this is not the case and usually such functions lack sufficient regularity if no additional assumptions are imposed. The reader is referred to [32,Example 7.5], where a function with the mean-value property is constructed for which 0 does not exist. Let be a minus fundamental solution of L on R (see Eq. 2.2 for definition). Note that in the case of the fractional Laplace operator for and some constant (see e.g. [10]). In what follows we will assume the kernel to satisfy the following growth condition: (G) 2 R 0 and there exists a non-increasing function 0 0 and 0 0 such that (i) if 1  The result uses quite involved conditions because the measure interacts with the Dinitype assumptions for the right-hand side function . Looking at examples, we see that the two cases described in the theorem appear naturally. In the fractional Laplacian case ( for ), finiteness of the expression 1 2 0 1 d depends on the value of 0 2 . We show in Section 6 that the conditions hold true when L is the generator of a rotationally symmetric -stable process, i.e., when L equals the fractional Laplace operator. Note that Theorem 1.2 is a new result even in this case. We also study the more general class, e.g. operators of the form , where is a Bernstein function. Note that in the theorems and remark above we do not assume that is bounded. Remark 1. 4 We emphasize that in the case of L being the fractional Laplace operator of order 0 2 and 2 1 , it is not true that the function 1 1 d belongs to 2 1 , or even to 1 1 1 as is stated in [1,Theorem 3.7]. A similar phenomenon has been mentioned in [3] and is visible here as well. Observe that in such case the integrals (1.5) and (1.6) are clearly divergent and consequently, Theorem 1.2 cannot be applied. We devote Section 5 to the construction of counterexamples for any 0 2 . On the other hand, by [39, Theorem 1.1(b)], those counterexamples are 1 1 1 for every 0 1 and consequently, they are also pointwise solutions. Thus, [1, Theorem 1.1] is also not true in general.
The article is organized as follows: in Section 2 we provide the main definitions and some preliminary results. The proof of Theorem 1.1 is provided in Section 3. Section 4 contains several rather technical computations and the proof of Theorem 1.2. We discuss the necessity of the assumptions of Theorem 1.2 through examples in Section 5. Finally, in Section 6 we provide examples that show that the assumptions of Theorem 1.2 are natural.

Preliminaries
In this section we explain our use of notation, define several objects and collect some basic facts. We write when and are comparable, that is the quotient stays between two positive constants. To simplify the notation, for a radial function we use the same symbol to denote its radial profile. In the whole paper and denote constants which may vary from line to line. We write when the constant depends only on . The family 0 induces a strongly continuous contraction semigroup on 0 R and 2 R R d R whose generator A has the Fourier symbol . Using the Kolmogorov theorem one can construct a stochastic process with transition densities , namely P d . Here P is the probability corresponding to a process starting from , that is P 0 1. By E we denote the corresponding expectation. In fact, is a pure-jump isotropic unimodal Lévy process in R , that is a stochastic process with stationary and independent increments and càdlàg paths, whose transition function is absolutely continuous and its density is isotropic unimodal, that is radial and radially non-increasing (see for instance [41] and [44]).
One of the objects of significant importance in this paper is the potential kernel defined as follows: 0 d .

Clearly
. The potential kernel can be defined in our setting if 1 1 . In particular, for 3 the potential kernel always exists (see [41,Theorem 37.8]). If this is not the case, one can consider the compensated potential kernel , we can set 0 0. In other cases the compensation must be taken with 0 R 0 . For details we refer the reader to [21] and to Appendix A.
Slightly abusing the notation, we let 1 be Eq. 2.1 for 0 0 ... 0 1 R . Thus, we have arrived with three potential kernels: , 0 and 1 . Each one corresponds to a different type of process and an operator associated with it. In order to merge these cases in one object, we let The basic object in the theory of stochastic processes is the first exit time of from , inf 0 .
Using we define an analogue of the generator of , namely, the characteristic operator or Dynkin operator. We say a Borel function is in a domain D U of Dynkin operator U if there exists a limit Here is understood as a limit over all sequences of open sets whose intersection is and whose diameters tend to 0 as . The characteristic operator is an extension For a wide description of characteristic operator and its relation with the generator of we refer the reader to [16,Chapter V].
Instead of the whole R , one can consider a process killed after exiting . By we denote its transition density (or, in other words, the fundamental solution of L in ). We have It follows that 0 . Proceeding as in the proof of [ Here we assume that the integral is absolutely convergent. If has the mean-value property in every bounded open set whose closure is contained in then is said to have the mean-value property inside .
Clearly if has the mean-value property inside , then U 0 in . In general, functions with the mean-value property lack sufficient regularity if no additional assumptions are imposed. In our setting, however, we can show that they are, in fact, twice continuously differentiable in . The proof is similar to the proof of [8,Theorem 4.6] and is omitted. [25,45] We say that a Borel function belongs to the Kato class K if it satisfies the following condition This is one of three conditions discussed by Zhao in [45]. A detailed description of different notions of the Kato class and related conditions can be found in [25]. R , it follows that 0 8 . Since 0 was arbitrary, the claim follows by induction.
A consequence of Lemma 2.6 is the following corollary.

Corollary 2.7 Let be open and bounded and
N. If R 0 1 loc then 1 .
The following lemma is crucial in one of the proofs. Proof Observe that thanks to Eq. 1.2 we get that L 1 for any 0. Now one may proceed exactly as in the proof of convergence of convolution in the classical 1 space.

Remark 2.10
Note that Lemma 2.9 fails without the assumption of doubling condition on . In fact, this condition is crucial even for the well-posedness of the problem, let alone further results.

Weak Solutions
The aim of this section is to prove Theorem 1.1. For the fractional Laplacian related results are known, cf. [7,Section 3]. A similar result has recently been obtained in [11] using purely analytic methods instead of probabilistic ones exploited in [7]. When the generalization of these results to more general nonlocal operators is immediate, we omit the proof. and consequently, L can be calculated pointwise. Now we proceed exactly as in the proof of [8,Lemma 4.10]. Note that we do not need the assumption A1 from [8], because it is used only to show that the considered function belongs to 2 ; furthermore, since the Lévy density is nonincreasing, it is continuous in a.e. 0 , so indeed the proof is exactly the same. Thus, the claim follows immediately.
The following lemma is a generalization of [7, Theorem 3.9 and Corollary 3.10], where the fractional Laplace operator is considered. R . By the strong Markov property we may assume that 1 is a Lipschitz domain. We claim that has the mean-value property in 1 . Indeed, let 2 be an open set relatively compact in such that 1 2 . There exist functions 1 , 2 on 1 such that 1 2 , 1 is continuous and bounded on 1 and 2 0 in 2 . We have The first integral is clearly absolutely convergent. We claim that it is also continuous as a function of in 1

The Sufficient Condition for Twice Differentiability
In this section, we provide auxiliary technical results and the proof of Theorem In particular, 4 3 for sufficiently small . It follows that , if is sufficiently small. Thus, is uniformly continuous. where the localization functions 1 and 2 are chosen in dependence of . Note that in the integral defining 2 , due to the function 2 and (G), integration w.r.t. takes place in a region where and its derivative are bounded. Hence, from (G) we see that differentiating under the integral sign is justified. We obtain 2 R

Counterexamples for the Case "α + β = 2"
In this section we provide several counterexamples for Theorem 1. where 0 2 . It is known (see [7] or Theorem 1. where is the (compensated) potential kernel for process whose generator is 2 . We claim that for any fixed 1 , the function E belongs to 1 . Indeed, using the explicit formula for the Poisson kernel for the unit ball [5], we get Observe that 1 , which implies that we are separated from the origin. It follows that 1 is locally bounded and since the integral converges at infinity for any form of , we obtain that for fixed 1

Case α ∈ (0, 1)
We follow closely the idea from the proof of Theorem 1.2 apart from the fact that at the end we will show that the last function 3 is not continuously differentiable. From Lemma 2.6 we get then the same argument applies for 4 . Therefore, it remains to calculate the derivative of 3 . Observe that on 1 4 we have 1 .
To simplify the notation we accept a mild ambiguity and by we denote, depending on the context, either a real number or a vector in R of the form 0 ... 0 . Let 0.  Observe that is equal to 0 for 0 and , thus, the expression in the first term of Eq. 5.8 simplifies. Furthermore, the second term vanishes, and the remaining limit is  We note that in our setting, is strictly decreasing.  Lemma A.2 allows us to introduce, following [5], [10], [28], a compensated potential kernel by setting for Observe that the left-hand side of Eq. A.5 converges to so it is finite. The remaining integral on the right-hand side converges as well by the monotone convergence theorem, but since all the other terms are finite, it follows that the integral is also finite and we obtain lim 0 E 0 E 0 which ends the proof.