Moser iteration applied to elliptic equations with critical growth on the boundary

TL;DR

This paper extends Moser iteration techniques to establish boundary boundedness and regularity of weak solutions for elliptic equations with critical growth functions, advancing understanding of boundary behavior in such problems.

Contribution

It introduces a modified Moser iteration method to prove boundary boundedness and regularity for elliptic equations with critical growth functions.

Findings

Weak solutions are bounded up to the boundary.

Under additional assumptions, solutions are $C^{1,eta}$-regular.

The method handles critical growth on the boundary.

Abstract

This paper deals with boundedness results for weak solutions of an elliptic equation where the functions are Carath\'eodory functions satisfying certain $p$-structure conditions that have critical growth even on the boundary. Based on a modified version of the Moser iteration we are able to prove that every weak solution of our problem is bounded up to the boundary. Under some additional assumptions this leads directly to $C^{1,\alpha}$-regularity for weak solutions of the problem.