Well-posed elliptic Neumann problems involving irregular data and domains

TL;DR

This paper investigates nonlinear elliptic Neumann problems in irregular domains with low integrability data, establishing existence, uniqueness, and stability of solutions using isocapacitary inequalities, with applications to diverse domain classes.

Contribution

It introduces new conditions linking data integrability and domain irregularity to ensure well-posedness of Neumann problems in complex settings.

Findings

Existence and uniqueness of solutions under low integrability conditions.

Continuous dependence of solutions on data.

Applicability to various irregular domains.

Abstract

Nonlinear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented.