𝐿_𝑝 regularity theorem for elliptic equations in less smooth domains

Abstract

We consider a 2𝑚th-order strongly elliptic operator 𝐴 subject to Dirichlet boundary conditions in a domain Ω of ℝ^𝑛, and show the 𝐿_𝑝 regularity theorem, assuming that the domain has less smooth boundary. We derive the regularity theorem from the following isomorphism theorems in Sobolev spaces. Let 𝑘 be a nonnegative integer. When 𝐴 is a divergence form elliptic operator, 𝐴 −𝜆 has a bounded inverse from the Sobolev space 𝑊_𝑝^{𝑘 −𝑚}(Ω) into 𝑊_𝑝^{𝑘 + 𝑚}(Ω) for 𝜆 belonging to a suitable sectorial region of the complex plane, if Ω is a uniformly 𝐶^𝑘,1 domain. When 𝐴 is a non-divergence form elliptic operator, 𝐴 −𝜆 has a bounded inverse from 𝑊_𝑝^𝑘(Ω) into 𝑊_𝑝^𝑘+2𝑚(Ω), if Ω is a uniformly 𝐶^𝑘+𝑚,1 domain. Compared with the known results, we weaken the smoothness assumption on the boundary of Ω by 𝑚 −1.