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Compactness for $GSBV^p$ via concentration-compactness

William M Feldman, Kerrek Stinson

TL;DR

This work addresses compactness for $GSBV^p$ functions without a priori bounds on the displacement, a key issue in fracture mechanics. It adopts Lions' concentration-compactness to reinterpret energy loss as concentration bubbles in the displacement range, yielding a partition of the domain into translator-induced bubbles and a vanishing core. The main contribution is a new, transparent compactness theorem that provides a subsequence, bubble centers, and a limit $u\in GSBV^p$ with precise convergence and jump-set properties, enabling existence results for nonlinear fracture energies. The approach connects concentration-compactness with free-discontinuity problems and offers a versatile framework potentially extendable to nonlinear finite elasticity and heterogeneous surface energies.

Abstract

Motivated by variational models for fracture, we provide a new proof of compactness for $GSBV^p$ functions without a priori bounds on the function itself. Our proof is based on the classical idea of concentration-compactness, making it transparent in strategy and simple in implementation. Further, so far as we are aware, this is the first time the connection to concentration-compactness has been made explicit for problems in fracture mechanics.

Compactness for $GSBV^p$ via concentration-compactness

TL;DR

This work addresses compactness for functions without a priori bounds on the displacement, a key issue in fracture mechanics. It adopts Lions' concentration-compactness to reinterpret energy loss as concentration bubbles in the displacement range, yielding a partition of the domain into translator-induced bubbles and a vanishing core. The main contribution is a new, transparent compactness theorem that provides a subsequence, bubble centers, and a limit with precise convergence and jump-set properties, enabling existence results for nonlinear fracture energies. The approach connects concentration-compactness with free-discontinuity problems and offers a versatile framework potentially extendable to nonlinear finite elasticity and heterogeneous surface energies.

Abstract

Motivated by variational models for fracture, we provide a new proof of compactness for functions without a priori bounds on the function itself. Our proof is based on the classical idea of concentration-compactness, making it transparent in strategy and simple in implementation. Further, so far as we are aware, this is the first time the connection to concentration-compactness has been made explicit for problems in fracture mechanics.

Paper Structure

This paper contains 4 sections, 5 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Proof for $GSBV^p$ compactness
  4. Lower semi-continuity of the jump-set

Key Result

Theorem 1.1

Let $1<p<\infty$, integers $d,N\geq 1$, and $\Omega\subset \Omega' \subset \mathbb{R}^N$ be Lipschitz domains. Let $(u_n)_{n\in \mathbb{N}} \in GSBV^p({\Omega'}; \mathbb{R}^d)$ be such that where $h\in W^{1,p}(\Omega';\mathbb{R}^d).$ Then there exists a subsequence of $(u_n)_{n\in \mathbb{N}}$ (not relabeled), a collection of disjoint sets of finite perimeter $\mathcal{S}_n:=(S_j^n)_{j=0}^{\infty

Figures (2)

  • Figure 1: Left: The material domain with crack $J_u$. Above right: The function $f(t;u)$ from \ref{['eqn:fn-intro']} measures concentrations in the range. Below right: The current configuration as deformed/displaced by $u(x)$.
  • Figure 2: Partitioning the range via the concentration-compactness bubbles. In the figure sets $E'$ denote $u(E)$, i.e. $P_1' = u(P_1)$ etc.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Remark 3.1
  • Remark 3.2: Example with vanishing
  • Lemma 3.3
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 3.4: Accounting for the partition
  • proof : Proof of Lemma \ref{['lem:vanishing']}
  • ...and 2 more