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Continuity up to the boundary for obstacle problems to porous medium type equations

Kristian Moring, Leah Schätzler

Abstract

We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This result seems to be new even for signed solutions to the (obstacle free) Cauchy-Dirichlet problem to the singular porous medium equation, which is retrieved as a special case.

Continuity up to the boundary for obstacle problems to porous medium type equations

Abstract

We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This result seems to be new even for signed solutions to the (obstacle free) Cauchy-Dirichlet problem to the singular porous medium equation, which is retrieved as a special case.
Paper Structure (28 sections, 25 theorems, 262 equations)

This paper contains 28 sections, 25 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. On the notion of solution
  4. Novelty and significance
  5. Definition and some properties of solutions
  6. Notation
  7. Definition of solutions
  8. Truncation of solutions to the obstacle problem
  9. Boundedness of solutions to the obstacle problem
  10. Auxiliary results
  11. Tools for local sub(super)solutions
  12. Tools for the case near zero
  13. Tools for the case away from zero
  14. Tools for sub(super)solutions near the initial boundary
  15. Tools for the case near zero
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $q \in (0,\infty)$ and $\Omega \subset \mathbb{R}^n$ be a bounded open set which satisfies the geometric density condition geometry for some $\alpha_*, \varrho_o >0$. Furthermore, suppose that the obstacle function $\psi$ satisfies eq:psi_conds, the lateral boundary datum $g$ satisfies eq:g-cond for every $(x_o,t_o) \in \mathcal{K} \cap S_T$ and $(x_1,t_1) \in \mathcal{K}$. Furthermore, if $K

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 36 more