# The ∞-Eigenvalue Problem

Research paper by Petri Juutinen, Peter Lindqvist, Juan J. Manfredi

Indexed on: 01 Sep '99Published on: 01 Sep '99Published in: Archive for Rational Mechanics and Analysis

#### Abstract

. The Euler‐Lagrange equation of the nonlinear Rayleigh quotient $$\left(\int_{\Omega}|\nabla u|^{p}\,dx\right) \bigg/ \left(\int_{\Omega}|u|^{p}\,dx\right)$$ is $$-\div\left( |\nabla u|^{p-2}\nabla u \right)= \Lambda_{p}^{p} |u |^{p-2}u,$$ where $$\Lambda_{p}^{p}$$ is the minimum value of the quotient. The limit as $$p\to\infty$$ of these equations is found to be $$\max \left\{ \Lambda_{\infty}-\frac{|\nabla u(x)|}{u(x)},\ \ \Delta_{\infty}u(x)\right\}=0,$$ where the constant $$\Lambda_{\infty}=\lim_{p\to\infty}\Lambda_{p}$$ is the reciprocal of the maximum of the distance to the boundary of the domain Ω.