Local and global regularity of weak solutions of elliptic equations with superquadratic Hamiltonian

TL;DR

This paper investigates the regularity of weak solutions to second-order elliptic equations with superquadratic gradient terms, demonstrating H"older continuity under certain conditions and highlighting the gradient's role in regularity.

Contribution

It establishes new regularity results for weak solutions of elliptic equations with superquadratic gradient growth, emphasizing the gradient term's influence.

Findings

Weak subsolutions are H"older continuous up to the boundary.

Local and global summability results are provided.

Gradient term is key to regularity, not the principal part.

Abstract

In this paper, we study the regularity of weak solutions and subsolutions of second-order elliptic equations having a gradient term with superquadratic growth. We show that, under appropriate integrability conditions on the data, all weak subsolutions in a bounded and regular open set $\Om$ are H\"older-continuous up to the boundary of $\Om$. Some local and global summability results are also presented. The main feature of this kind of problems is that the gradient term, not the principal part of the operator, is responsible for the regularity.