Maximal solutions for the Infinity-eigenvalue problem

Abstract

In this article we prove that the first eigenvalue of the $\infty -$Laplacian $$\{\begin{array}{rclclcccc}\min \{-{\Delta }_{\infty }v,|\nabla v|-{\lambda }_{1,\infty }(\Omega )v\}& =& 0& \text{in}& \Omega v& =& 0& \text{on}& \partial \Omega ,\end{array}$$ has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as $\ell \nearrow 1$ of concave problems of the form $$\{\begin{array}{rclclcccc}\min \{-{\Delta }_{\infty }{v}_{\ell },|\nabla v_{\ell }|-{\lambda }_{1,\infty }(\Omega )v_{\ell }^{\ell }\}& =& 0& \text{in}& \Omega v_{\ell }& =& 0& \text{on}& \partial \Omega .\end{array}$$ In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the concave problems as happens for the usual eigenvalue problem for the $p-$Laplacian for a fixed $1<p<\infty$.

14 pages

References in corpus (3)

• The \infty eigenvalue problem and a problem of optimal transportation
• Rigidity results for variational infinity ground states
• On the geometry of the $p$-Laplacian operator