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Characterizing BV- and BD-ellipticity for a class of positively 1-homogeneous surface energy densities

Dominik Engl, Carolin Kreisbeck, Marco Morandotti

Abstract

Lower semicontinuity of surface energies in integral form is known to be equivalent to BV-ellipticity of the surface density. In this paper, we prove that BV-ellipticity coincides with the simpler notion of biconvexity for a class of densities that depend only on the jump height and jump normal, and are positively 1-homogeneous in the first argument. The second main result is the analogous statement in the setting of bounded deformations, where we show that BD-ellipticity reduces to symmetric biconvexity. Our techniques are primarily inspired by constructions from the analysis of structured deformations and the general theory of free discontinuity problems.

Characterizing BV- and BD-ellipticity for a class of positively 1-homogeneous surface energy densities

Abstract

Lower semicontinuity of surface energies in integral form is known to be equivalent to BV-ellipticity of the surface density. In this paper, we prove that BV-ellipticity coincides with the simpler notion of biconvexity for a class of densities that depend only on the jump height and jump normal, and are positively 1-homogeneous in the first argument. The second main result is the analogous statement in the setting of bounded deformations, where we show that BD-ellipticity reduces to symmetric biconvexity. Our techniques are primarily inspired by constructions from the analysis of structured deformations and the general theory of free discontinuity problems.
Paper Structure (10 sections, 14 theorems, 102 equations, 3 figures)

This paper contains 10 sections, 14 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Auxiliary results
  5. $\rm BV$-ellipticity and related notions
  6. Basic definitions and properties
  7. Characterization results
  8. $\rm BD$-ellipticity and related notions
  9. Basic definitions
  10. Characterization result

Key Result

Theorem 1.1

If $f\colon{\mathbb{R}}^n\times{\mathcal{S}}^{n-1}\to [0,\infty)$ is even and positively $1$-homogeneous in the first variable, then $f$ is $\rm BV$-elliptic if and only if $f$ is biconvex.

Figures (3)

  • Figure 1: (a) the function $\varphi_k$ and (b) the function $u_k$ in dimension $n=2$ (here pictured for $k=4$), the dashed lines marking the jump set $J_{u_k}$. For the purpose of illustration, here we have taken $\lambda,\xi\in{\mathbb{R}}^2$ with vanishing first component.
  • Figure 2: An illustration of the triangle $\Delta_k$, including $k^2-1$ parallel lines with normal $\lambda$ that intersect the bottom line equidistantly. All other lengths are uniquely determined as a consequence of the intercept theorem.
  • Figure 3: An illustration of the translated copies $\Delta_k^i$ of $\Delta_k$.

Theorems & Definitions (36)

  • Theorem 1.1: Characterization of $\rm \bm{BV}$-ellipticity
  • Theorem 1.2: Characterization of $\rm \bm{BD}$-ellipticity
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3: Alternative representations of $\bm{\Phi_f}$
  • Lemma 2.4
  • Definition 3.1: $\rm \bm{BV}$-ellipticity
  • Definition 3.2: Biconvexity
  • ...and 26 more