Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fine boundary continuity for degenerate double-phase diffusion

Simone Ciani, Eurica Henriques, Igor Skrypnik

Abstract

We study the boundary behavior of solutions to parabolic double-phase equations through the celebrated Wiener's sufficiency criterion. The analysis is conducted for cylindrical domains and the regularity up to the lateral boundary is shown in terms of either its $p$ or $q$ capacity, depending on whether the phase vanishes at the boundary or not. Eventually we obtain a fine boundary estimate that, when considering uniform geometric conditions as density or fatness, leads us to the boundary Hölder continuity of solutions. In particular, the double-phase elicits new questions on the definition of an adapted capacity.

Fine boundary continuity for degenerate double-phase diffusion

Abstract

We study the boundary behavior of solutions to parabolic double-phase equations through the celebrated Wiener's sufficiency criterion. The analysis is conducted for cylindrical domains and the regularity up to the lateral boundary is shown in terms of either its or capacity, depending on whether the phase vanishes at the boundary or not. Eventually we obtain a fine boundary estimate that, when considering uniform geometric conditions as density or fatness, leads us to the boundary Hölder continuity of solutions. In particular, the double-phase elicits new questions on the definition of an adapted capacity.
Paper Structure (23 sections, 13 theorems, 200 equations, 2 figures)

This paper contains 23 sections, 13 theorems, 200 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Origins and Framing of the Topic
  3. Fine Boundary Regularity
  4. Setting of the Problem
  5. Main Result and Applications
  6. Notation
  7. Preliminaries
  8. Definition of solution
  9. Local Energy Estimates and Critical Mass Lemma
  10. Testing with negative powers towards a Reverse Hölder's inequality
  11. Weak Harnack's Inequality
  12. Geometric Setting and Auxiliary Results
  13. Preamble
  14. Geometric setting
  15. Capacity estimates
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $(x_o,t_o) \in S_T$ and let $u$ be a bounded, weak solution to the Cauchy-Dirichlet problem C-pb. Depending on the point $(x_o,t_o)$, we assume that either or Then, in each case respectively, there exist $\{\rho_0(p),\eta_0(p)\}$, $\{ \rho_0(q), \eta_0(q)\}$ couples of positive numbers depending only on the data and conditions eq1.6-eq1.7, and positive constants $\gamma, \hat{\gamma}, \gamm

Figures (2)

  • Figure 1: Scheme of the geometric setting of the proof. For the definition of $\eta^*,\eta_*$ see Subsection \ref{['geometric setting']} below. Considered a same radius $r$, when $a(x_o,t_o)$ approaches zero, $\eta_*$ stretches to infinity while $\eta^*$ stays unvaried. This motivates the reduction of radii $r<R$ in the former case, according to the size of the phase.
  • Figure 2: Comparing time lengths in proof of Lemma \ref{['lem3.3']}.

Theorems & Definitions (25)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 3.1: Energy Estimates
  • Lemma 3.2: Initial-values Critical Mass
  • proof
  • Remark 3.3
  • Corollary 3.4
  • ...and 15 more