First-order evolution equations with dynamic boundary conditions.

Abstract

In this paper, we introduce a general framework to study linear first-order evolution equations on a Banach space X with dynamic boundary conditions, that is with boundary conditions containing time derivatives. Our method is based on the existence of an abstract Dirichlet operator and yields finally to equivalent systems of two simpler independent equations. In particular, we are led to an abstract Cauchy problem governed by an abstract Dirichlet-to-Neumann operator on the boundary space ∂X. Our approach is illustrated by several examples and various generalizations are indicated. This article is part of the theme issue 'Semigroup applications everywhere'.