Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1

Anna Mercaldo,Julio D Rossi,Cristina Trombetti,Sergio Segura De León

doi:10.3934/cpaa.2013.12.253

Abstract

We consider the solution up to the Neumann problem for the p{ Laplacian equation with the normal component of the ∞ux across the boundary given by g 2 L 1 (@›). We study the behaviour of up as p goes to 1 showing that they converge to a measurable function u and the gradients jrupj pi2 rup converge to a vector fleld z. We prove that z is bounded and that the properties of u depend on the size of g measured in a suitable norm: if g is small enough, then u is a function of bounded variation (it vanishes on the whole domain, when g is very small) while if g is large enough, then u takes the value 1 on a set of positive measure. We also prove that in the flrst case, u is a solution to a limit problem that involves the 1iLaplacian. Finally, explicit examples are shown.

Highlights

In this paper we deal with the limit as p goes to 1 of solutions to the p–Laplacian with non-homogeneous Neumann boundary conditions
If we argue formally the limit limp→1 up = u should be a solution to the following limit problem that involves the
Note that one of the major difficulties to define a solution to this problem is to give a sense to when Du = 0. This difficulty was tackled for the equation in (1.4) with Dirichlet boundary conditions and a nontrivial right hand side, see

Summary

Introduction

In this paper we deal with the limit as p goes to 1 of solutions to the p–Laplacian with non-homogeneous Neumann boundary conditions. Consider the following problem: −div |∇up|p−2∇up = 0,. Note that one of the major difficulties to define a solution to this problem is to give a sense to when Du = 0 This difficulty was tackled for the equation in (1.4) with Dirichlet boundary conditions and a nontrivial right hand side, see [10], [12], [13] and the book [3]. The behaviour of the solutions up is studied, we prove that up converge to a measurable function u whose main features depend on the size of g.

Notation and auxiliary results

Convergence of up as p goes to 1

Existence of solutions to the limit problem

Examples