Sobolev regularity theory for the non-local elliptic and parabolic equations on $C^{1,1}$ open sets

TL;DR

This paper establishes Sobolev regularity results for non-local elliptic and parabolic equations involving fractional Laplacians on smooth bounded domains, providing existence, uniqueness, and regularity estimates in weighted Sobolev spaces.

Contribution

It introduces sharp weighted Sobolev and Hölder regularity estimates for solutions of fractional elliptic and parabolic equations on $C^{1,1}$ domains, extending the theory to arbitrary derivative orders.

Findings

Proves existence and uniqueness of solutions in weighted Sobolev spaces.

Derives global Sobolev and Hölder estimates for solutions and derivatives.

Identifies sharp range of weights related to boundary distance for regularity.

Abstract

We study the zero exterior problem for the elliptic equation $$ \Delta^{\alpha/2}u-\lambda u=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $\alpha\in (0,2)$, $\lambda \geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and H\"older estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and H\"older regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.