Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

TL;DR

This paper proves the boundedness of solutions to various boundary value problems for elliptic equations with nonlinearities governed by general convex functions in Orlicz spaces, extending classical results to critical growth conditions.

Contribution

It introduces new boundedness results for elliptic boundary value problems with nonlinearities in Orlicz spaces, utilizing optimal embedding theorems for the first time.

Findings

Solutions are globally bounded under Dirichlet, Neumann, and Robin boundary conditions.

Optimal Orlicz-Sobolev and trace embeddings are crucial for handling critical growth.

Results extend classical boundedness to more general nonlinearities in Orlicz spaces.

Abstract

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.