[HTML][HTML] Boundary regularity for nonlocal operators with kernels of variable orders
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 L u= 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∈ CV (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).
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 L u= 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∈ CV (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).
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 L u= 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).
Elsevier
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