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Two-phase free boundary problems for operators with nonstandard growth

Fausto Ferrari, Monica Jacob, Claudia Lederman

Abstract

In this paper we focus the attention on free boundary problems ruled by partial differential equations with nonstandard growth, presenting in particular some recent results. The interest in these problems stems from the diverse applications that motivate their study and from the challenging mathematical difficulties they pose.

Two-phase free boundary problems for operators with nonstandard growth

Abstract

In this paper we focus the attention on free boundary problems ruled by partial differential equations with nonstandard growth, presenting in particular some recent results. The interest in these problems stems from the diverse applications that motivate their study and from the challenging mathematical difficulties they pose.
Paper Structure (22 sections, 15 theorems, 88 equations)

This paper contains 22 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Some models with variable exponents
  3. Image processing
  4. Non-Newtonian fluids
  5. Introduction
  6. Electrorheological fluids
  7. Thermorheological fluids
  8. Non-homogeneous porous media
  9. Thermistors modeling
  10. Some basics on free boundary problems
  11. Nonstandard growth equations: some basic tools
  12. One-phase free boundary problems for nonstandard growth operators
  13. Assumptions
  14. Basic definitions, notation and preliminaries
  15. Regularity results for viscosity solutions and their free boundaries
  16. ...and 7 more sections

Key Result

Theorem 5.2

Let $p$ and $f$ be as in Definition defnweak. Assume moreover that $f\in C(\Omega)$ and $p\in C^1(\Omega)$. Let $u\in W^{1,p(\cdot)}(\Omega)\cap C(\Omega)$ be a weak solution to $\Delta_{p(x)}u=f$ in $\Omega$. Then $u$ is a viscosity solution to $\Delta_{p(x)}u=f$ in $\Omega$.

Theorems & Definitions (19)

  • Definition 5.1
  • Theorem 5.2: Theorem 3.2 in FL1
  • Definition 5.3
  • Definition 5.4
  • Theorem 5.5
  • Theorem 5.6: Optimal regularity, Theorem 1.1 in FL2
  • Theorem 5.7: Flatness implies $C^{1,\alpha}$, Theorem 1.1 in FL1
  • Theorem 5.8: Lipschitz implies $C^{1,\alpha}$, Theorem 1.2 in FL2
  • Proposition 5.9
  • Definition 6.1
  • ...and 9 more