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Variational convergences under moving anisotropies

Alberto Maione, Fabio Paronetto, Simone Verzellesi

TL;DR

This work develops a comprehensive variational framework for sequences of moving anisotropies, providing Gamma-convergence results for local functionals $F_h(u)=\int_\Omega f_h(x,X^h u)\,dx$ and identifying their limits as functionals of the limit anisotropy $X$. It introduces uniform approximation tools (Meyers–Serrin, affine-space approximations, Anzellotti–Giaquinta type results) and uniform compactness and Poincaré inequalities that accommodate nonuniform coercivity across the sequence. A central contribution is the Gamma-compactness theorem for moving anisotropies, including the convergence of norms, Dirichlet-boundary problems, and quadratic forms, along with the convergence of minima, minimizers, and momenta. These results culminate in an H-convergence theory for moving anisotropies, showing that anisotropic linear operators with evolving coefficients admit a limit operator and convergent solutions, thus bridging static and moving anisotropy analysis in sub-Riemannian/Hörmander-type contexts.

Abstract

We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincaré inequalities and compactness properties. We prove several $Γ$-convergence results, and apply the latter to the study of $H$-convergence of anisotropic linear differential operators.

Variational convergences under moving anisotropies

TL;DR

This work develops a comprehensive variational framework for sequences of moving anisotropies, providing Gamma-convergence results for local functionals and identifying their limits as functionals of the limit anisotropy . It introduces uniform approximation tools (Meyers–Serrin, affine-space approximations, Anzellotti–Giaquinta type results) and uniform compactness and Poincaré inequalities that accommodate nonuniform coercivity across the sequence. A central contribution is the Gamma-compactness theorem for moving anisotropies, including the convergence of norms, Dirichlet-boundary problems, and quadratic forms, along with the convergence of minima, minimizers, and momenta. These results culminate in an H-convergence theory for moving anisotropies, showing that anisotropic linear operators with evolving coefficients admit a limit operator and convergent solutions, thus bridging static and moving anisotropy analysis in sub-Riemannian/Hörmander-type contexts.

Abstract

We study the asymptotic behaviour of sequences of integral functionals depending on moving anisotropies. We introduce and describe the relevant functional setting, establishing uniform Meyers-Serrin type approximations, Poincaré inequalities and compactness properties. We prove several -convergence results, and apply the latter to the study of -convergence of anisotropic linear differential operators.

Paper Structure

This paper contains 18 sections, 22 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Anisotropic functional frameworks
  3. Anisotropies
  4. Sobolev spaces
  5. Moving anisotropies
  6. Preliminary functional properties
  7. Uniform approximation by smooth functions
  8. A uniform Meyers-Serrin type approximation
  9. A uniform characterization of affine Sobolev spaces
  10. A uniform Anzellotti-Giaquinta type approximation
  11. Uniform compactness and Poincaré inequalities
  12. Gamma-convergence for functionals depending on moving anisotropies
  13. Gamma-convergence of norms
  14. Gamma-compactness in the general case
  15. Gamma-convergence for functionals with Dirichlet boundary conditions
  16. ...and 3 more sections

Key Result

Theorem 2.4

For any $1\leqslant p <\infty$, it holds that $H^{1,p}_X(\Omega)=W^{1,p}_X(\Omega).$

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5: Poincaré inequality
  • Theorem 2.6: Rellich-Kondrachov
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 51 more