Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain

TL;DR

This paper proves that solutions to the Neumann problem for the Poisson equation in convex domains are uniformly Lipschitz, enhancing understanding of solution regularity in such geometries.

Contribution

It establishes the uniform Lipschitz continuity of solutions in convex domains, extending regularity results to arbitrary convex n-dimensional spaces.

Findings

Solutions are uniformly Lipschitz in convex domains.

Application to regularity of solutions on convex polyhedra.

Improved understanding of boundary behavior in Neumann problems.

Abstract

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given.