Semigroups Generated by Elliptic Operators in Non-Divergence Form on $C_0(/omega)$

TL;DR

This paper proves that certain second-order elliptic operators generate holomorphic semigroups on continuous functions vanishing at the boundary of Lipschitz domains, extending previous results to less regular domains.

Contribution

It establishes generation of holomorphic semigroups by elliptic operators with continuous coefficients on Lipschitz domains, relaxing regularity assumptions.

Findings

Elliptic operators generate holomorphic semigroups on C_0(omega).

Results apply to Lipschitz domains, not just smoother ones.

Dirichlet problem is also addressed for these operators.

Abstract

Given a bounded domain in the Euclidean space satisfying the uniform outer cone condition, we show that a uniformly elliptic operator of second order with continuous second order coefficients generates a holomorphic semigroup on the space of all continuous functions vanishing at the boundary. In particular, Lipschitz domains are allowed. This result was so far known under considerable stronger regularity assumptions. Also the Dirichlet problem is considered for such operators.