Nonlinear Maximal Monotone Extensions of Symmetric Operators

Given a linear semi-bounded symmetric operator $S\ge -ω$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_Θ$ of type $λ>ω$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $λ$) which coincide with the Friedrichs extension of $S$ on a convex set containing ${\mathscr D}(S)$. The extension parameter $Θ\subset{\mathfrak h}\times{\mathfrak h}$ ranges over the set of nonlinear maximal monotone relations on an auxiliary Hilbert space $\mathfrak h$ isomorphic to the deficiency subspace of $S$. Moreover $A_Θ+λ$ is a sub-potential operator (i.e. is the sub-differential of a lower semicontinuos convex function) whenever $Θ$ 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.

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