The limit as p→+∞p→+∞ of the first eigenvalue for the pp-Laplacian with mixed Dirichlet and Robin boundary conditions

We analyze the behavior as p→∞p→∞ of the first eigenvalue of the pp-Laplacian with mixed boundary conditions of Dirichlet–Robin type. We find a nontrivial limit that we associate to a variational principle involving L∞L∞-norms. Moreover, we provide a geometrical characterization of the limit value as well as a description of it using optimal mass transportation techniques. Our results interpolate between the pure Dirichlet case and the mixed Dirichlet–Neumann case.