Nonlinear Maximal Monotone Extensions of Symmetric Operators

Abstract

Given a linear semi-bounded symmetric operator $S\ge -\omega$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_\Theta$ of type $\lambda>\omega$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $\lambda$) which coincide with the Friedrichs extension of $S$ on a convex set containing ${\mathscr D}(S)$. The extension parameter $\Theta\subset{\mathfrak h}\times{\mathfrak h}$ ranges over the set of nonlinear maximal monotone relations on an auxiliary Hilbert space $\mathfrak h$ isomorphic to the deficiency subspace of $S$. Moreover $A_\Theta+\lambda$ is a sub-potential operator (i.e. is the sub-differential of a lower semicontinuos convex function) whenever $\Theta$ is sub-potential. Examples describing Laplacians with nonlinear singular perturbations supported on null sets and Laplacians with nonlinear boundary conditions on a bounded set are given.