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

Abstract

We study the zero exterior problem for the elliptic equation $${\Delta }^{\alpha /2}u-\lambda u=f,x\in D;u{|}_{D^c}=0$$ as well as for the parabolic equation $$u_t={\Delta }^{\alpha /2}u+f,t>0,x\in D;u(0,\cdot ){|}_D=u_0,u{|}_{[0,T]\times D^c}=0.$$ Here, $\alpha \in (0,2)$, $\lambda \ge 0$ 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 Hölder estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and Hölder 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.

arXiv comment42 pages; Title changed; The paper has been accepted for publication in "Discrete and Continuous Dynamical Systems (DCDS)"