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Superlinear free-discontinuity models: relaxation and phase field approximation

Sergio Conti, Matteo Focardi, Flaviana Iurlano

TL;DR

This work extends the Direct Method in the Calculus of Variations to superlinear free-discontinuity energies defined on $SBV$, where jump sets may have infinite measure. It decouples and analyzes the surface and bulk contributions to establish lower semicontinuity, provides a relaxation result for the full energy without strictly assuming quasiconvexity or BV-ellipticity, and introduces phase-field approximations via anisotropic Ambrosio–Tortorelli functionals with rigorous $\Gamma$-convergence to a cohesive-type limit. The phase-field analysis introduces two surface-energy densities $g_{\inf}$ and $g_{\sup}$, proves their equality under a projection property of the $q$-recession $\Psi_{\infty}$, and yields a practical variational framework for numerically approximating fracture-type energies with small-jump (cohesive) effects. Collectively, the results advance the mathematical understanding of relaxation and numerical approximation for vector-valued, superlinear free-discontinuity problems with potentially infinite fracture sets, with implications for fracture mechanics and related multi-field phase-field models.

Abstract

In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on $SBV$ functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase field models.

Superlinear free-discontinuity models: relaxation and phase field approximation

TL;DR

This work extends the Direct Method in the Calculus of Variations to superlinear free-discontinuity energies defined on , where jump sets may have infinite measure. It decouples and analyzes the surface and bulk contributions to establish lower semicontinuity, provides a relaxation result for the full energy without strictly assuming quasiconvexity or BV-ellipticity, and introduces phase-field approximations via anisotropic Ambrosio–Tortorelli functionals with rigorous -convergence to a cohesive-type limit. The phase-field analysis introduces two surface-energy densities and , proves their equality under a projection property of the -recession , and yields a practical variational framework for numerically approximating fracture-type energies with small-jump (cohesive) effects. Collectively, the results advance the mathematical understanding of relaxation and numerical approximation for vector-valued, superlinear free-discontinuity problems with potentially infinite fracture sets, with implications for fracture mechanics and related multi-field phase-field models.

Abstract

In this paper we develop the Direct Method in the Calculus of Variations for free-discontinuity energies whose bulk and surface densities exhibit superlinear growth, respectively for large gradients and small jump amplitudes. A distinctive feature of this kind of models is that the functionals are defined on functions whose jump sets may have infinite measure. Establishing general lower semicontinuity and relaxation results in this setting requires new analytical techniques. In addition, we propose a variational approximation of certain superlinear energies via phase field models.
Paper Structure (13 sections, 23 theorems, 381 equations, 1 figure)

This paper contains 13 sections, 23 theorems, 381 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Lower semicontinuity
  4. Lower semicontinuity for surface energies
  5. Lower semicontinuity for bulk energies
  6. Lower semicontinuity for free-discontinuity energies
  7. Relaxation
  8. Phase field approximation
  9. Model
  10. Characterizations of the surface energy densities
  11. Structural properties of the surface energy densities
  12. $\Gamma$-liminf
  13. $\Gamma$-limsup

Key Result

Proposition 1.1

Let $g_0$ and $g$ satisfy gBVell-e:g0 superlinear, and let ${u_j}\in {(GSBV(\Omega))^m}$ satisfy Then

Figures (1)

  • Figure 1: Sketch of the construction in the proof of Theorem \ref{['relaxation2']}. The set $\Omega$ (blue boundary) is covered by finitely many simplexes $E_k$ (light blue); a large part of the interior of each of them is contained in the sets $E_k'$. The codimension-1 sets $F_j'$ (red) cover most of the part of the faces inside $\Omega$, and are in turn covered by the union of small cubes of size $\rho$ (green).

Theorems & Definitions (42)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • proof : Proof of Proposition \ref{['p:lsc surface']}
  • Proposition 2.3
  • ...and 32 more