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Characterizations and properties of solutions to parabolic problems of linear growth

Theo Elenius

TL;DR

This work develops a rigorous framework for parabolic problems with linear growth in the BV setting, where the natural unknowns are measure-valued gradients. It introduces two time-regularity-free notions of weak solutions: variational solutions, defined by a BV-energy inequality with boundary terms, and weak solutions via the Anzellotti pairing, possibly extended by a time-integrated formulation. The paper proves the equivalence of these notions (under appropriate regularity) and provides a stability mechanism that handles 1-homogeneous functionals by approximating with differentiable functionals. It also establishes a comparison principle ensuring uniqueness of variational solutions, along with approximation results (parabolic SBV approximation and time mollification) and a local boundedness result achieved without a priori bounds. The results yield a robust framework for analyzing regularity and dynamics of parabolic problems with linear growth, including the total variation flow as a canonical example, and provide tools for further study of BV-type parabolic equations.

Abstract

We consider notions of weak solutions to a general class of parabolic problems of linear growth, formulated independently of time regularity. Equivalence with variational solutions is established using a stability result for weak solutions. A key tool in our arguments is approximation of parabolic BV functions using time mollification and Sobolev approximations. We also prove a comparison principle and a local boundedness result for solutions. When the time derivative of the solution is in $L^2$ our definitions are equivalent with the definition based on the Anzellotti pairing.

Characterizations and properties of solutions to parabolic problems of linear growth

TL;DR

This work develops a rigorous framework for parabolic problems with linear growth in the BV setting, where the natural unknowns are measure-valued gradients. It introduces two time-regularity-free notions of weak solutions: variational solutions, defined by a BV-energy inequality with boundary terms, and weak solutions via the Anzellotti pairing, possibly extended by a time-integrated formulation. The paper proves the equivalence of these notions (under appropriate regularity) and provides a stability mechanism that handles 1-homogeneous functionals by approximating with differentiable functionals. It also establishes a comparison principle ensuring uniqueness of variational solutions, along with approximation results (parabolic SBV approximation and time mollification) and a local boundedness result achieved without a priori bounds. The results yield a robust framework for analyzing regularity and dynamics of parabolic problems with linear growth, including the total variation flow as a canonical example, and provide tools for further study of BV-type parabolic equations.

Abstract

We consider notions of weak solutions to a general class of parabolic problems of linear growth, formulated independently of time regularity. Equivalence with variational solutions is established using a stability result for weak solutions. A key tool in our arguments is approximation of parabolic BV functions using time mollification and Sobolev approximations. We also prove a comparison principle and a local boundedness result for solutions. When the time derivative of the solution is in our definitions are equivalent with the definition based on the Anzellotti pairing.

Paper Structure

This paper contains 15 sections, 26 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functions of bounded variation
  4. Function of a measure
  5. Parabolic function spaces
  6. Definition of variational and weak solutions
  7. Approximation results
  8. Approximation with parabolic Sobolev functions
  9. Mollification in time
  10. Comparison principle
  11. Weak solutions
  12. Differentiable functionals
  13. Anzellotti-type Euler equations
  14. Stability and equivalence
  15. Local boundedness of solutions

Key Result

Lemma 2.1

Let $u\in \mathop{\mathrm{BV}}\nolimits(\Omega)$ and $v\in \mathop{\mathrm{BV}}\nolimits(\mathbb R^n\backslash\overline{\Omega})$. Then the function belongs to $\mathop{\mathrm{BV}}\nolimits(\mathbb R^n)$, and its distributional gradient is given by where $\nu_\Omega$ denotes the generalized inner unit normal to $\Omega$. In the above we interpret $Du$ and $Dv$ as measures on the whole of $\math

Theorems & Definitions (50)

  • Lemma 2.1: AmbrosioFuscoPallara
  • Lemma 2.2: CDMLP1988
  • Proposition 2.3: Anzellotti:1984
  • Lemma 2.4
  • Lemma 2.5
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • Lemma 2.8: BoegelDuzSchevObstacle:2016
  • Definition 2.9
  • ...and 40 more