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Regularity of Supersolutions

Peter Lindqvist

Abstract

The regularity for the supersolutions of the Evolutionary p-Laplace Equation is considered. In particular,the equivalence of viscosity supersolutions and p-supercaloric functions (lower semicontinuous supersolutions defined via a comparison principle) is considered. Bounded viscosity supersolutions are, in fact, weak supersolutions belonging to a natural Sobolev Space.

Regularity of Supersolutions

Abstract

The regularity for the supersolutions of the Evolutionary p-Laplace Equation is considered. In particular,the equivalence of viscosity supersolutions and p-supercaloric functions (lower semicontinuous supersolutions defined via a comparison principle) is considered. Bounded viscosity supersolutions are, in fact, weak supersolutions belonging to a natural Sobolev Space.

Paper Structure

This paper contains 17 sections, 48 theorems, 391 equations.

Table of Contents

  1. Introduction
  2. The Stationary Equation
  3. The Evolutionary Equation
  4. Definitions
  5. A Separable Minorant.
  6. Bounded Supersolutions
  7. Harnack's Convergence Theorem
  8. Unbounded Supersolutions
  9. Proof of the Theorem
  10. Weak Supersolutions Are Semicontinuous
  11. The Equation With Measure Data
  12. Pointwise Behaviour
  13. The Stationary Equation
  14. The Evolutionary Equation
  15. Viscosity Supersolutions Are Weak Supersolutions
  16. ...and 2 more sections

Key Result

Proposition 1

Suppose that $v$ is a superharmonic function defined in $\mathbb{R}^{n}$. Then the Sobolev derivative $\nabla v$ exists and whenever $0 < q < \frac{n}{n-1}$. Moreover, for $\eta \geq 0,\,\eta \in C_{0}^{\infty}(\mathbb{R}^{n})$.

Theorems & Definitions (54)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 5
  • Lemma 6: Comparison Principle
  • Lemma 7: Harnack's Inequality
  • Definition 8
  • Definition 9
  • Theorem 10
  • ...and 44 more