Dynamical boundary conditions in a non-cylindrical domain for the Laplace equation

TL;DR

This paper investigates the mathematical properties of the Laplace equation with dynamic boundary conditions on non-cylindrical domains, using advanced operator methods to analyze existence, uniqueness, and asymptotic behavior.

Contribution

It introduces a novel framework employing non-autonomous Dirichlet-to-Neumann operators to handle dynamical boundary conditions on complex domains.

Findings

Proves existence and uniqueness of solutions.

Analyzes asymptotic behavior of solutions.

Extends methods to non-autonomous elliptic problems.

Abstract

In this paper, we study existence, uniqueness and asymptotic behavior of the Laplace equation with dynamical boundary conditions on regular non-cylindrical domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator and use form methods in a more general framework to accomplish our goal. A class of non-autonomous elliptic problems with dynamical boundary conditions on Lipschitz domains is also considered in this same context.