Boundary regularity for nonlocal operators with kernels of variable orders

Abstract

We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of Lu = f in D, u = 0 in R n \ D, where D is a bounded C 1,1 open set, and that the solution u satisfies u ∈ C V (D) and u/V (d D ) ∈ C α (D) with the uniform estimates, where V is the renewal function and d D (x) = dist(x, ∂D).

On overdetermined problems for a general class of nonlocal operators

“…Note that u > 0 in D by the maximum principle. Moreover, as explained above, u is Hölder continuous in R d [21] and thus in particular bounded. Given λ ∈ R, e ∈ ∂B 1 (0) denote v(x) = v λ,e (x) = u(x) − u(x), x ∈ R d , wherex := R λ,e (x) := x − 2(x · e)e + 2λe denotes the reflection of x at T λ,e := ∂H λ,e , H λ,e := {x ∈ R d : x · e > λ}.…”

“…where in the last step we used [20, Proposition 3.1] (see also [21,Lemma 2.5]). Combining this with (3.7) the claim follows.…”

“…(1.4) There is large family of subordinators that satisfy (1.3) (see [5,21] For some of our proofs below we use some information on the normalized ascending ladder-height process of {X 1 t } t≥0 , where X 1 t denotes the first coordinate of X t . Recall that the ascending ladderheight process of a Lévy process Z is the process of the right inverse (Z L −1 t ) t≥0 , where L t is the local time of Z t reflected at its supremum (for details and further information we refer to [3,Chapter 6]).…”

Remarks on the Nonlocal Dirichlet Problem

“…Note that, in the case where L equals the fractional Laplace operator, similar results like Theorem 1.1 are proved in [6]. A result similar to Theorem 1.1 has recently been proved in [26]. The authors consider a smaller class of operators and concentrate on viscosity solutions instead of distributional solutions.…”

Hopf’s lemma for viscosity solutions to a class of non-local equations with applications

“…See also [20] for a strong maximum principle for a class of operators using a different approach. As it is well known, see [39],…”