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Non-autonomous overdetermined problems for the normalized p-Laplacian

Lucio Cadeddu, Antonio Greco, Benyam Mebrate

Abstract

We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition at the boundary. The definitions and statements needed to understand the main results are recalled in detail.

Non-autonomous overdetermined problems for the normalized p-Laplacian

Abstract

We present existence and nonexistence results on the solution of an overdetermined problem for the normalized p-Laplacian in a bounded open set, with p ranging from 1 to infinity. More precisely we consider a non-constant Neumann condition at the boundary. The definitions and statements needed to understand the main results are recalled in detail.
Paper Structure (4 sections, 7 theorems, 32 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Viscosity Solutions
  3. Well-posedness, comparison principle, radial solutions
  4. Existence and nonexistence of solutions

Key Result

Theorem 1.1

Let $p \in [1,\infty]$ and let $c_p$ be as above.

Figures (1)

  • Figure 1: Sternberg & Ziemer's example.

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Example 1.4
  • Remark 1
  • Definition 2.1
  • Definition 2.2
  • Remark 2
  • Lemma 3.1: Comparison principle
  • proof
  • ...and 11 more