[Paper Review] An abstract approach to domain perturbation for parabolic equations and parabolic variational inequalities

Parinya Sa Ngiamsunthorn|arXiv (Cornell University)|Sep 15, 2011

Advanced Mathematical Modeling in Engineering15 references3 citations

TL;DR

This paper establishes an abstract framework for analyzing domain perturbation in non-autonomous parabolic equations and variational inequalities using Mosco convergence. It proves that Mosco convergence of function spaces for parabolic problems is equivalent to that of corresponding elliptic problems, enabling convergence of solutions under domain changes without requiring smooth domains or change of variables, with applications to Dirichlet and Neumann problems and parabolic variational inequalities.

ABSTRACT

We study the behaviour of solutions of linear non-autonomous parabolic equations subject to Dirichlet or Neumann boundary conditions under perturbation of the domain. We prove that Mosco convergence of function spaces for non-autonomous parabolic problems is equivalent to Mosco convergence of function spaces for the corresponding elliptic problems. As a consequence, we obtain convergence of solutions of non-autonomous parabolic equations under domain perturbation by variational methods using the same characterisation of domains as in elliptic case. A similar technique can be applied to obtain convergence of weak solutions of parabolic variational inequalities when the underlying convex set is perturbed.

Motivation & Objective

• To establish a general abstract framework for studying the convergence of solutions to non-autonomous parabolic equations under domain perturbation.
• To address the challenge of singular domain perturbations—such as dumbbell domains or domains with cracks—where standard change-of-variable techniques fail.
• To extend the theory of Mosco convergence from elliptic to parabolic problems, proving equivalence between parabolic and elliptic Mosco convergence for function spaces.
• To prove convergence of weak solutions for parabolic variational inequalities when the underlying convex set is perturbed.
• To provide a unified variational method applicable to both Dirichlet and Neumann boundary conditions, even when domains lack regularity.

Proposed method

• Formulates non-autonomous parabolic equations in abstract evolution form in Bochner–Sobolev spaces $ W((0,T),V,V') $ and Bochner–Lebesgue spaces $ L^2((0,T),V) $.
• Applies Mosco convergence of closed, convex subsets in Banach spaces to characterize convergence of solutions under domain perturbation.
• Proves equivalence between Mosco convergence of function spaces for parabolic problems and their corresponding elliptic counterparts via Theorem 3.4.
• Uses variational methods and coercivity estimates to derive weak and strong convergence of solutions in $ L^2((0,T),V) $.
• Constructs approximating solutions $ u_{ ho,n} $ via time-regularization (e.g., $ ho u' + u = u $) to satisfy Mosco condition (M1') and pass to the limit.
• Employs compactness and weak lower semicontinuity arguments to show convergence of solutions under assumptions on data convergence and Mosco convergence of domains.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of a non-autonomous parabolic equation converge when the domain undergoes perturbation?
• RQ2 How can Mosco convergence of function spaces be used to characterize solution convergence in parabolic problems?
• RQ3 Is the equivalence between Mosco convergence of parabolic and elliptic problems valid for both Dirichlet and Neumann boundary conditions?
• RQ4 Can the same abstract framework be extended to parabolic variational inequalities with perturbed convex sets?
• RQ5 What are the minimal regularity assumptions on domains for solution convergence in the Neumann case, especially when no smooth extension exists?

Key findings

• Mosco convergence of function spaces for non-autonomous parabolic problems is equivalent to Mosco convergence for the corresponding elliptic problems, as established in Theorem 3.4.
• Solutions of non-autonomous parabolic equations with Dirichlet or Neumann boundary conditions converge weakly in $ L^2((0,T),V) $ under Mosco convergence of domains and data convergence.
• Strong convergence of solutions in $ L^2((0,T),V) $ is achieved when $ f_n o f $ in $ L^2((0,T),V') $, $ u_{0,n} ightharpoonup u_0 $ in $ V $, and $ u_{0,n} o u_0 $ in $ H $, under Mosco convergence of domains.
• The method applies to both Dirichlet and Neumann problems, even when domains are irregular and no smooth extension of $ H^1( heta_n) $ to $ H^1( heta) $ exists.
• The framework extends to parabolic variational inequalities: weak solutions converge when the underlying convex set $ K_n $ converges to $ K $ in the sense of Mosco.
• The proof technique avoids reliance on compactness results from [8, Lemma 2.1], which fail in the Neumann case, by using abstract Mosco convergence and regularization.