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Variational properties of local functionals driven by arbitrary anisotropies

Simone Verzellesi

Abstract

We provide integral representation and $Γ$-compactness results for anisotropic local functionals depending on arbitrary Lipschitz continuous vector fields. In particular, neither bracket-generating assumptions nor linear independence conditions are required.

Variational properties of local functionals driven by arbitrary anisotropies

Abstract

We provide integral representation and -compactness results for anisotropic local functionals depending on arbitrary Lipschitz continuous vector fields. In particular, neither bracket-generating assumptions nor linear independence conditions are required.
Paper Structure (15 sections, 11 theorems, 63 equations)

This paper contains 15 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and motivations
  3. Main notation
  4. Lipschitz continuous vector fields
  5. Relevant vector fields
  6. $\Gamma$-convergence and local functionals
  7. Anisotropic representation of Euclidean Lagrangians
  8. Algebraic properties of the Moore-Penrose pseudo-inverse
  9. The anisotropic representation result
  10. Integral representation and $\Gamma$-compactness without \ref{['lic']}: proofs in a prototypical example
  11. Integral representation
  12. $\Gamma$-compactness
  13. Further statements
  14. Integral representation
  15. $\Gamma$-compactness

Key Result

Proposition 3.1

Let $\mathcal{C}_{P}$ be the above-defined map. Moreover, for any $x\in\Omega$, let $\mathcal{C}_{P}(x):{\mathbb{R}}^m\longrightarrow {\mathbb{R}}^n$ be the linear map defined by for any $\eta$. Then the map is measurable for any $\eta\in{\mathbb{R}}^m$. Moreover, for any $x\in\Omega$, the following facts hold.

Theorems & Definitions (19)

  • Example 2.1
  • Example 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Example 4.1
  • Theorem 4.2
  • ...and 9 more