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Hölder regularity for the linearized porous medium equation in bounded domains

Tianling Jin, Jingang Xiong

Abstract

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the Hölder regularity of their weak solutions.

Hölder regularity for the linearized porous medium equation in bounded domains

Abstract

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the Hölder regularity of their weak solutions.
Paper Structure (17 sections, 36 theorems, 346 equations)

This paper contains 17 sections, 36 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. Sobolev spaces and inequalities
  3. Some weighted Sobolev spaces
  4. Sobolev inequalities
  5. Weak solutions
  6. Definitions
  7. Energy estimates, uniqueness and existence
  8. $W^{1,1}_2$ regularity
  9. Boundedness of weak solutions
  10. A maximum principle
  11. A local maximum principle
  12. Hölder regularity
  13. Improvement of oscillations centered at the boundary
  14. Interior Hölder estimates
  15. Hölder estimates near the boundary
  16. ...and 2 more sections

Key Result

Lemma 2.1

For $p>0$, $\mathring W^{1,1}_2(Q^+_{R,T})$ is dense in $\mathring V^{1,1}_2(Q^+_{R,T})$.

Theorems & Definitions (72)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5: Hardy's inequality
  • Lemma 2.6
  • ...and 62 more