Mathematical analysis of a parabolic-elliptic problem with moving parabolic subdomain through a Lagrangian approach

Abstract

The aim of this paper is to study the regularity of the solution of some linear parabolic-elliptic problems in which parabolicity region depends on time. More specifically, this region is the position occupied by a body undergoing a motion (a deformation smoothly evolving in time). The main tool we introduce is a suitable extension of the motion to the entire spatial domain of the PDE. This enables us to reduce the original problem to a parabolic-elliptic problem with variable coefficients and with a parabolicity region independent of time. This problem can be seen as a Lagrangian formulation of our original problem. Next, we obtain regularity results for a class of parabolic-elliptic problems with variable coefficients and fixed parabolicity region. We apply these results to the Lagrangian formulation and, finally, we obtain a regularity result for our original problem.