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Relaxation of quasi-convex functionals with variable exponent growth

Giacomo Bertazzoni, Petteri Harjulehto, Peter Hästö, Elvira Zappale

TL;DR

This work analyzes the relaxation of quasi-convex bulk functionals with variable exponent growth in BV-type spaces, proving an integral representation for the lower semicontinuous envelope in BV^{p(\cdot)} when the exponent p is log-Hölder continuous. The authors decompose the energy into an absolutely continuous part and a singular part via the recession function, capturing linear and super-linear growth regimes and, crucially, the case p(x)=1. The main contribution is a rigorous relaxation framework that handles mixed growth and anisotropy, providing a compelling model for materials with nonstandard growth and potential jumps in the relaxed energy. The results rely on approximation by linear-growth functionals and a blow-up method to obtain Radon measure representations, ensuring lower semicontinuity and an explicit integral form.

Abstract

We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function.

Relaxation of quasi-convex functionals with variable exponent growth

TL;DR

This work analyzes the relaxation of quasi-convex bulk functionals with variable exponent growth in BV-type spaces, proving an integral representation for the lower semicontinuous envelope in BV^{p(\cdot)} when the exponent p is log-Hölder continuous. The authors decompose the energy into an absolutely continuous part and a singular part via the recession function, capturing linear and super-linear growth regimes and, crucially, the case p(x)=1. The main contribution is a rigorous relaxation framework that handles mixed growth and anisotropy, providing a compelling model for materials with nonstandard growth and potential jumps in the relaxed energy. The results rely on approximation by linear-growth functionals and a blow-up method to obtain Radon measure representations, ensuring lower semicontinuity and an explicit integral form.

Abstract

We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function.

Paper Structure

This paper contains 4 sections, 13 theorems, 110 equations.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. BV-type spaces
  4. Proof of main result

Key Result

Theorem 1.3

Let $\Omega$ be a bounded open set with Lipschitz boundary, $p: \Omega\to [1,\infty)$ be $\log$-Hölder continuous and assume that $f:\mathbb R^{m\times n}\to [0,\infty)$ satisfies the following assumptions. Then the relaxed functional has the integral representation for all $u \in {\rm BV}^{p(\cdot)}(\Omega; {\mathbb R}^m)$.

Theorems & Definitions (29)

  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Weak lower semicontinuity
  • Lemma 2.5: Reshetnyak continuity, Theorem 2.38, AmbFP00
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 19 more