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Critical points of the one dimensional Ambrosio-Tortorelli functional with an obstacle condition

Martin Rakovsky

Abstract

We consider a family of critical points of the Ambrosio-Tortorelli energy with an obstacle condition on the phase field variable. This problem can be interpreted as a time discretization of a quasistatic evolution problem where the obstacle at step $n$ is defined as the solution at step $n-1$. The obstacle condition now reads as an irreversibility condition (the crack can only increase in time). The questions tackled here are the regularity of the critical points, the properties inherited from the obstacle sequence, the position of the limit points and the equipartition of the phase field energy. The limits of such critical points turn out to be critical points of the Mumford-Shah energy that inherit the possible discontinuities induced by the obstacle sequence.

Critical points of the one dimensional Ambrosio-Tortorelli functional with an obstacle condition

Abstract

We consider a family of critical points of the Ambrosio-Tortorelli energy with an obstacle condition on the phase field variable. This problem can be interpreted as a time discretization of a quasistatic evolution problem where the obstacle at step is defined as the solution at step . The obstacle condition now reads as an irreversibility condition (the crack can only increase in time). The questions tackled here are the regularity of the critical points, the properties inherited from the obstacle sequence, the position of the limit points and the equipartition of the phase field energy. The limits of such critical points turn out to be critical points of the Mumford-Shah energy that inherit the possible discontinuities induced by the obstacle sequence.

Paper Structure

This paper contains 20 sections, 32 theorems, 228 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and statement of the results
  3. Notations and preliminary results
  4. Euler-Lagrange equations
  5. Statement of the theorems
  6. Critical points of $\overline{AT}_\varepsilon$
  7. $C^{1,1}$ regularity of the critical points
  8. Shape of $v_\varepsilon$
  9. Asymptotic analysis
  10. Preliminary estimates
  11. Estimates on $v_\varepsilon$
  12. Possible values for the limit slope
  13. Convergence of the Dirichlet energy
  14. Convergence of the phase-field term
  15. Equipartition of the energy
  16. ...and 5 more sections

Key Result

theorem 2.1

Up to a subsequence, $(u_{0,\varepsilon} , v_{0,\varepsilon}) \rightarrow (u_0, 1)$ in $L^2(0,L)^2$, where either $u_0=u_{jump} \equiv a_0 1_{[L/2,L]}$ or $u_0(x)=u_{aff}(x) = a_0x/L$.

Figures (2)

  • Figure 1: Possible values of $\Gamma_0$, $J_{u_1}$ and $\Gamma_1$.
  • Figure 2: Critical points of $v_\varepsilon$

Theorems & Definitions (70)

  • definition 1
  • theorem 2.1
  • lemma 1
  • definition 2
  • theorem 2.2
  • theorem 2.3: Convergence of the critical points
  • remark 2.1
  • theorem 2.4: $\Gamma$ convergence of $\overline{AT}_\varepsilon$
  • remark 2.2
  • remark 2.3
  • ...and 60 more