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The inhomogeneous Total Variation Flow with $L^1$-data

Marta Latorre, Sergio Segura de León

Abstract

This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the $1$--Laplacian operator under minimal integrability assumptions. Specifically, we consider \begin{equation*} u'-\Div(Du/|D u|)=f\qquad\text{ in } (0,+\infty)\timesΩ\,, \end{equation*} where $Ω\subset\R^N$ is a bounded open set with Lipschitz boundary, $u_0\in L^1(Ω)$ is the initial datum, and $f\in L_{loc}^1(0,+\infty; L^1(Ω))$ is the source term. We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the \mbox{$1$--Laplacian} structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case.

The inhomogeneous Total Variation Flow with $L^1$-data

Abstract

This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the --Laplacian operator under minimal integrability assumptions. Specifically, we consider \begin{equation*} u'-\Div(Du/|D u|)=f\qquad\text{ in } (0,+\infty)\timesΩ\,, \end{equation*} where is a bounded open set with Lipschitz boundary, is the initial datum, and is the source term. We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the \mbox{--Laplacian} structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case.
Paper Structure (9 sections, 13 theorems, 131 equations)

This paper contains 9 sections, 13 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular data framework
  4. An auxiliary result
  5. Entropy solution
  6. Uniqueness
  7. Comparison between solutions
  8. Regularity
  9. Long term decay for the homogeneous problem

Key Result

Proposition 3.4

For every $f\in L^2((0,T)\times \Omega)$ and every $u_0\in W^{1,1}(\Omega) \cap L^2(\Omega)$, problem P admits a unique solution in the sense of Definition def-reg.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Lemma 3.6
  • Proposition 3.7
  • Remark 3.8
  • ...and 13 more