Boundary Lipschitz regularity of solutions for semilinear elliptic equations in divergence form

TL;DR

This paper establishes boundary Lipschitz regularity for solutions of semilinear elliptic equations in divergence form under weaker boundary and data assumptions, extending to Reifenberg C^{1,Dini} domains.

Contribution

It proves boundary Lipschitz regularity under minimal conditions on the boundary and data, generalizing previous results to Reifenberg C^{1,Dini} domains.

Findings

Solutions are Lipschitz continuous at boundary points under specified conditions.

Results extend regularity to Reifenberg C^{1,Dini} domains.

Weaker assumptions on the nonhomogeneous term and boundary are sufficient.

Abstract

In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.