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A Harnack inequality for solutions of elliptic-parabolic equations

Fabio Paronetto

TL;DR

This work develops a Harnack-type theory for strongly degenerate parabolic and elliptic-parabolic equations with a density $\uprho(x,t) \ge 0$, by introducing a parabolic De Giorgi class of order $2$, constructing an inhomogeneous parabolic Harnack for approximants, and passing to the limit to obtain a Harnack inequality for solutions in the degenerate regime. The analysis relies on an expansion of positivity, delicate energy estimates, and advanced covering and weight-theory tools to handle regions where $\uprho$ vanishes. A key outcome is Hölder continuity across the interface where $\uprho$ changes sign, and the results are extended to elliptic-parabolic mixtures by considering the elliptic behavior in the vanishing-density zones. The results have potential applications to diffusion and flow problems in heterogeneous media with phase interfaces.

Abstract

We want to prove a Harnack type inequality for solutions of strongly degenerate parabolic, or elliptic-parabolic, equations. To do that, we first define a De Giorgi class of order $p = 2$ that contains the solutions of evolution equations of the types $\uprho (x,t) u_t + A u = 0$ and $(\uprho (x,t) u)_t + A u = 0$, where $\uprho > 0$ almost everywhere and $A$ is a suitable elliptic operator. For functions belonging to this class we prove an inhomogeneous parabolic Harnack inequality, i.e. a Harnack inequality that takes into account the mean value of $\uprho$ in different regions of $Ω\times (0,T)$. \\ As a consequence, thanks to an approximation result and a delicate passage to the limit, we are able to get a Harnack inequality for solutions, and in these cases only for solutions, of strongly degenerating parabolic equations, i.e. when $\uprho \geqslant 0$. \\ As a byproduct one obtains Hölder continuity for solutions of a subclass of the first equation (i.e. $\uprho (x,t) u_t + A u = 0$): in particular the solutions of this subclass are Hölder continuous in the interface where $\uprho$ changes its sign, from positive to zero.

A Harnack inequality for solutions of elliptic-parabolic equations

TL;DR

This work develops a Harnack-type theory for strongly degenerate parabolic and elliptic-parabolic equations with a density , by introducing a parabolic De Giorgi class of order , constructing an inhomogeneous parabolic Harnack for approximants, and passing to the limit to obtain a Harnack inequality for solutions in the degenerate regime. The analysis relies on an expansion of positivity, delicate energy estimates, and advanced covering and weight-theory tools to handle regions where vanishes. A key outcome is Hölder continuity across the interface where changes sign, and the results are extended to elliptic-parabolic mixtures by considering the elliptic behavior in the vanishing-density zones. The results have potential applications to diffusion and flow problems in heterogeneous media with phase interfaces.

Abstract

We want to prove a Harnack type inequality for solutions of strongly degenerate parabolic, or elliptic-parabolic, equations. To do that, we first define a De Giorgi class of order that contains the solutions of evolution equations of the types and , where almost everywhere and is a suitable elliptic operator. For functions belonging to this class we prove an inhomogeneous parabolic Harnack inequality, i.e. a Harnack inequality that takes into account the mean value of in different regions of . \\ As a consequence, thanks to an approximation result and a delicate passage to the limit, we are able to get a Harnack inequality for solutions, and in these cases only for solutions, of strongly degenerating parabolic equations, i.e. when . \\ As a byproduct one obtains Hölder continuity for solutions of a subclass of the first equation (i.e. ): in particular the solutions of this subclass are Hölder continuous in the interface where changes its sign, from positive to zero.
Paper Structure (8 sections, 15 theorems, 336 equations)

This paper contains 8 sections, 15 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, notations and assumptions
  3. DG classes
  4. Expansion of positivity
  5. The Harnack inequality
  6. A preliminary result
  7. A Harnack inequality for solutions of elliptic-parabolic equations
  8. Hölder continuity

Key Result

Theorem 2.5

Consider $q > 2$, $r > 0$, $x_o \in {\bf R}^n$, $\omega \in L^{\infty} ({\bf R}^n)$, $\omega > 0$ a.e., $\omega \in B_{2,q}^{1} (K_1)$ and doubling, i.e. satisfying sob-poin-cond and doubling. Then there is a constant $\tilde{\upgamma}$ depending $($only$)$ on $n, q, K_1, c_{\omega}$ such that for every $u$ Lipschitz continuous function defined in $B_{r} = B_{r}(x_0)$, with either support contain

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.8
  • Theorem 2.9
  • Lemma 2.10
  • Lemma 2.11
  • Definition 3.1
  • Proposition 4.1
  • Proposition 4.3
  • ...and 7 more