On the Solution of Boundary Value Problems Set in Domains With Moving Boundaries

Abstract

We construct solutions for time-dependent boundary value problems set in moving domains with Dirichlet, Neumann, and mixed boundary conditions. When the boundaries are time deformations of an initial boundary along a vector field, we can refer the boundary problem to a fixed domain at the cost of increasing the complexity of the coefficients. This strategy works well for heat equations under general boundary conditions. However, it leads to hyperbolic problems including damping terms of the form ∇𝑢𝑡 for wave equations, which we are able to solve with zero Dirichlet boundary conditions. For more general boundaries, extension techniques leading to measure valued sources allow us to construct solutions for heat problems with Neumann boundary conditions.