Archive for Rational Mechanics and Analysis

, Volume 148, Issue 2, pp 89-105

The ∞-Eigenvalue Problem

  • Petri JuutinenAffiliated withDepartment of Mathematics, University of Jyväskylä, 40351 Jyväskylä, Finland
  • , Peter LindqvistAffiliated withDepartment of Mathematics, Norwegian Institute of Technology, Trondheim N-7034, Norway
  • , Juan J. ManfrediAffiliated withDepartment of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

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. 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.