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 ${\mathbb{R}}^n\setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u\in C^V(D)$ and $u/V(d_D)\in C^{\alpha }(D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.

References in corpus (7)

• An extension problem related to the fractional Laplacian
• Regularity theory for fully nonlinear integro-differential equations
• Boundary regularity for fully nonlinear integro-differential equations
• Dirichlet Heat Kernel Estimates for Rotationally Symmetric Lévy processes
• Schauder estimates in generalized Hölder spaces
• Geometric property of the Ground State Eigenfunction for Cauchy Process
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Cited by in corpus (4)

• Local Hölder regularity for nonlocal equations with variable powers
• Existence and non-existence results for a class of semilinear nonlocal operators with exterior condition
• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights
• On overdetermind problems for a general class of nonlocal operators