Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space

TL;DR

This paper establishes existence and regularity results for nonlinear Robin boundary-value problems involving harmonic and polyharmonic functions in half-spaces, considering general growth conditions in Orlicz spaces.

Contribution

It introduces new sharp boundedness properties in Orlicz spaces for classical harmonic analysis operators, enabling analysis of nonlinear boundary conditions.

Findings

Existence and regularity of solutions for nonlinear Robin problems.

Extension to Orlicz growth conditions including power, logarithmic, and exponential types.

Development of new boundedness results in Orlicz spaces for harmonic analysis operators.

Abstract

Existence and global regularity results for boundary-value problems of Robin type for harmonic and polyharmonic functions in $n$-dimensional half-spaces are offered. The Robin condition on the normal derivative on the boundary of the half-space is prescribed by a nonlinear function $\mathcal N$ of the relevant harmonic or polyharmonic functions. General Orlicz type growths for the function $\mathcal N$ are considered. For instance, functions $\mathcal N$ of classical power type, their perturbations by logarithmic factors, and exponential functions are allowed. New sharp boundedness properties in Orlicz spaces of some classical operators from harmonic analysis, of independent interest, are critical for our approach.