Global boundedness of weak solutions with finite energy to a general class of Dirichlet problems

TL;DR

This paper establishes global boundedness results for weak solutions with finite energy to a broad class of nonlinear elliptic Dirichlet problems, extending regularity theory beyond minimizers to more general solutions.

Contribution

It introduces new boundedness results applicable to weak solutions with finite energy for general nonlinear elliptic equations, not limited to minimizers.

Findings

Proves global boundedness under general growth conditions.

Extends regularity results to broader class of weak solutions.

Applicable to various cases in recent PDE literature.

Abstract

As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness results under general assumptions that can be applied to several cases studied in the recent and extensive literature on partial differential equations \textit{under general growth}. In particular, we propose the class of \textit{weak solutions with finite energy} in which to search for solutions and in which regularity can be studied and achieved. We emphasize that we are not limited to minimizers of certain integral functionals, as often considered recently in this context of general growth, but to the broader class of weak solutions to Dirichlet problems for general nonlinear elliptic equations in divergence form.