##### Sesión Ecuaciones Diferenciales y Probabilidad

Diciembre 14, 11:00 ~ 11:20

## Maximal solutions for a fully nonlinear eigenvalue problem

### da Silva, João Vitor

In this talk we show that the first eigenvalue of the following (fully nonlinear) $\infty-$eigenvalue problem $$\left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v,\, |\nabla v|-\lambda_{1, \infty}(\Omega) v \} & = & 0 & \text{in} & \Omega \\ v & = & 0 & \text{on} & \partial \Omega, \end{array} \right.$$ has a unique (up to scalar multiplication) maximal viscosity solution. Such a solution is obtained as the limit as $\ell \nearrow 1$ of a family of solutions of concave problems as follows $$\left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v_{\ell},\, |\nabla v_{\ell}|-\lambda_{1, \infty}(\Omega) v_{\ell}^{\ell} \} & = & 0 & \text{in} & \Omega \\ v_{\ell} & = & 0 & \text{on} & \partial \Omega, \end{array} \right.$$ which will enable us to conclude 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

Autores: da Silva, João Vitor / Rossi, Julio / Salort, Ariel.