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’.

中文翻译：

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具有动态边界条件的一阶演化方程

在本文中，我们引入了一个通用框架来研究具有动态边界条件的 Banach 空间 X 上的线性一阶演化方程，即包含时间导数的边界条件。我们的方法基于抽象狄利克雷算子的存在，最终产生两个更简单的独立方程的等效系统。特别是，我们被引向了一个抽象的柯西问题，该问题由边界空间 ∂X 上的抽象 Dirichlet-to-Neumann 算子控制。我们的方法通过几个例子来说明，并指出了各种概括。这篇文章是主题问题“无处不在的半组应用程序”的一部分。