Laplace problem with an exponential nonlinear boundary condition

TL;DR

This paper proves existence and uniqueness of solutions for the Laplace equation with exponential Robin boundary conditions on the unit disk, using an iterative method and Sobolev embeddings.

Contribution

It introduces a new existence and uniqueness result for Laplace problems with exponential nonlinear boundary conditions on the disk.

Findings

Existence and uniqueness of solutions under small boundary data

Application of iterative method with Sobolev embeddings

Results specific to exponential Robin boundary conditions

Abstract

In this paper, we establish a new result for the Laplace problem with exponential Robin boundary conditions posed on the unit disk in $\R^2$. More precisely, we prove the existence and uniqueness of a solution under suitable smallness assumptions on the boundary data. Our approach relies on an iterative method combined with periodic Sobolev embedding results.