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Total $\mathbb{A}$-variation flows

David Meyer

Abstract

We study the $L^2$-gradient flows, $\partial_t u-\mathrm{div}(\mathrm{D}f(x,\mathbb{A}u))=0$, of functionals of the type $\int_Ωf(x,\mathbb{A}u)\,\mathrm{d}x$, where $f$ is a convex function of linear growth and $\mathbb{A}$ is some first-order linear constant-coefficient differential operator. To this end, we identify the relaxation of the functional to the space $\mathrm{BV}^{\mathbb{A}}\cap L^2$, identify its subdifferential, and show pointwise representation formulas for the relaxation and the subdifferential, both with and without Dirichlet boundary conditions. The existence and uniqueness then follow from abstract semigroup theory. We further show that our solutions can be obtained as limits of the corresponding flows with $p$-growth as $p\searrow 1$.

Total $\mathbb{A}$-variation flows

Abstract

We study the -gradient flows, , of functionals of the type , where is a convex function of linear growth and is some first-order linear constant-coefficient differential operator. To this end, we identify the relaxation of the functional to the space , identify its subdifferential, and show pointwise representation formulas for the relaxation and the subdifferential, both with and without Dirichlet boundary conditions. The existence and uniqueness then follow from abstract semigroup theory. We further show that our solutions can be obtained as limits of the corresponding flows with -growth as .
Paper Structure (11 sections, 23 theorems, 56 equations)

This paper contains 11 sections, 23 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Existing literature
  3. Structure of the paper
  4. Preliminaries and Notation
  5. General notation
  6. The space $\mathop{\mathrm{BV^\mathbb{A}}}\nolimits$
  7. Preliminaries from convex analysis
  8. Duality pairings for $\mathop{\mathrm{BV^\mathbb{A}}}\nolimits$ and Green's formula
  9. Green's formula
  10. Main results for Neumann boundary conditions
  11. The functional with Dirichlet boundary conditions

Key Result

Corollary 1.3

There is a unique solution to main eq for initial data in $L^2(\Omega,\mathbb{R}^n)$, which can be described with the subdifferential characterisation above.

Theorems & Definitions (44)

  • Corollary 1.3
  • Definition 2.1
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • Lemma 2.9
  • ...and 34 more