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Existence theory for linear-growth variational integrals with signed measure data

Eleonora Ficola, Thomas Schmidt

TL;DR

This work develops a semicontinuity-based BV existence theory for scalar linear-growth variational functionals with signed measure data. By embedding the problem in a cylinder via an extra variable and using a perspective extension of the integrand, the authors reduce to an anisotropic total variation framework and apply Reshetnyak-type results under isoperimetric conditions tied to the recession $f^{\infty}$. They prove lower semicontinuity under ICs with constant 1, establish existence of minimizers when the IC constant $C$ is strictly less than 1, and exhibit a sharp nonexistence example at the borderline $C=1$; they also develop a recovery-sequence construction and relate the BV-relaxation to the $W^{1,1}$-infimum. The results significantly extend area-type and anisotropic total-variation theories to general linear-growth functionals with measure data and provide a variational path toward the corresponding Euler–Lagrange equations with right-hand side $\mu$.

Abstract

We develop a semicontinuity-based existence theory in $\mathrm{BV}$ for a general class of scalar linear-growth variational integrals with additional signed-measure terms. The results extend and refine previous considerations for anisotropic total variations and area-type cases, and they pave the way for a variational approach to the corresponding Euler-Lagrange equations, which involve the signed measure as right-hand-side datum.

Existence theory for linear-growth variational integrals with signed measure data

TL;DR

This work develops a semicontinuity-based BV existence theory for scalar linear-growth variational functionals with signed measure data. By embedding the problem in a cylinder via an extra variable and using a perspective extension of the integrand, the authors reduce to an anisotropic total variation framework and apply Reshetnyak-type results under isoperimetric conditions tied to the recession . They prove lower semicontinuity under ICs with constant 1, establish existence of minimizers when the IC constant is strictly less than 1, and exhibit a sharp nonexistence example at the borderline ; they also develop a recovery-sequence construction and relate the BV-relaxation to the -infimum. The results significantly extend area-type and anisotropic total-variation theories to general linear-growth functionals with measure data and provide a variational path toward the corresponding Euler–Lagrange equations with right-hand side .

Abstract

We develop a semicontinuity-based existence theory in for a general class of scalar linear-growth variational integrals with additional signed-measure terms. The results extend and refine previous considerations for anisotropic total variations and area-type cases, and they pave the way for a variational approach to the corresponding Euler-Lagrange equations, which involve the signed measure as right-hand-side datum.
Paper Structure (19 sections, 28 theorems, 132 equations)

This paper contains 19 sections, 28 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation, measures, $\mathrm{BV}$ functions, and sets of finite perimeter
  4. Polars of positively 1-homogeneous integrands
  5. Anisotropic total variations and anisotropic perimeters
  6. Linear-growth integrands, recession and perspective function
  7. General convex functionals of measures
  8. Product structure and slicing of ${\mathcal{H}}^N$
  9. Statement of the main results
  10. Lower semicontinuity and existence theory
  11. The functional with extra variable
  12. ICs with extra variable
  13. Lower semicontinuity
  14. Coercivity and existence
  15. An example of non-existence in case of the borderline IC
  16. ...and 4 more sections

Key Result

Lemma 2.2

Consider a positively $1$-homoge-neous function $\varphi\colon {\mathds{R}}^N \to [0,\infty)$ such that $\varphi(\xi)>0$ holds for all $\xi\in{\mathds{R}}^N\setminus\{0\}$. Then, for every differentiability point $\xi^\ast\in{\mathds{R}}^N\setminus\{0\}$ of $\varphi^\circ$, the corresponding $\xi\in holds for every differentiability point $\xi^\ast\in{\mathds{R}}^N\setminus\{0\}$ of $\varphi^\circ

Theorems & Definitions (71)

  • Definition 2.1: polar
  • Lemma 2.2: equality cases in the anisotropic Cauchy-Schwarz inequality
  • proof
  • Definition 2.3: anisotropic total variation and anisotropic perimeter
  • Theorem 2.4: Reshetnyak semicontinuity; homogeneous version
  • Theorem 2.5: Reshetnyak continuity; homogeneous version
  • Theorem 2.6: $\varphi$-anisotropic isoperimetric inequality
  • Definition 2.7
  • Lemma 2.8
  • proof : Proof of Lemma \ref{['lem:properties_rec_persp']}
  • ...and 61 more