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

TL;DR

This paper proves boundary Lipschitz regularity for solutions of general semilinear elliptic equations in divergence form under weaker conditions on coefficients and data, using optimal assumptions like Dini continuity.

Contribution

It establishes boundary Lipschitz regularity under weaker and optimal conditions, including Dini continuity and Lipschitz Newtonian potential, for a broad class of semilinear elliptic equations.

Findings

Boundary Lipschitz regularity holds under weaker conditions.

Optimal conditions include Dini continuity and Lipschitz Newtonian potential.

Results apply to general semilinear elliptic equations in divergence form.

Abstract

In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.