Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A fully nonlinear transmission problem degenerating on the interface

Davide Giovagnoli, David Jesus

Abstract

In this paper we prove that solutions to a transmission problem degenerating on the interface are Hölder differentiable up to the interface with universal estimates. Furthermore, we obtain a sharper pointwise $C^{1,α(\cdot)}$ with optimal variable exponent and uniform estimates.

A fully nonlinear transmission problem degenerating on the interface

Abstract

In this paper we prove that solutions to a transmission problem degenerating on the interface are Hölder differentiable up to the interface with universal estimates. Furthermore, we obtain a sharper pointwise with optimal variable exponent and uniform estimates.

Paper Structure

This paper contains 14 sections, 23 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy of the results.
  4. Preliminaries
  5. Aleksandrov-Bakelman-Pucci estimate
  6. Harnack inequality and Hölder regularity
  7. Flat interface problems
  8. Viscosity solutions
  9. Lower and upper $\varepsilon$-envelopes
  10. Comparison principle and uniqueness
  11. A stability result for $C^2$ interfaces
  12. Approximation result for $C^{1,\alpha_0}$ interfaces
  13. Gradient regularity
  14. Acknowledgements

Key Result

Theorem 2.1

Suppose that the assumptions Assumption1-Assumption3 are in force and let $u \in C(B_1)$ be a viscosity solution to the transmission problem eq:main. Then $u \in C^{1, \alpha}(\overline{\Omega_{1/2}^{\pm}})$ with $\alpha = \min \{\alpha_0^-, 1- \bar{a}\}$ where $\Omega_{1/2}^\pm := \Omega^\pm \cap B where $p=d(1+1/\bar{a})/2>d$ and $C$ universal depending only on $d, \lambda, \Lambda, \bar{a}$ and

Theorems & Definitions (41)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1: Viscosity solution
  • Definition 3.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Remark 1
  • Theorem 4.1: ABP
  • Lemma 4.1
  • ...and 31 more