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A class of highly degenerate elliptic operators: maximum principle and unusual phenomena

Banff International Research Station for Mathematical Innovation and Discovery

We discuss the validity of the maximum principle below the principal eigenvalue for viscosity solutions of the Dirichlet problem in bounded domains $$ {\cal P}^-_{k}(D^2u)+H(x,\nabla u)+\mu u=0\quad\mbox{in}\;\Omega,\quad u=0\quad\mbox{on}\;\partial \Omega, $$ where the higher order term is given by the truncated Laplacian $ {\cal P}^-_{k}(D^2u)=\sum_{i=1}^k\lambda_i(D^2u), $ being $\lambda_i(D^2u)$ being the ordered eigenvalues of the Hessian. Some very unusual phenomena due to the degeneracy of the operator will be emphasized by means of explicit counterexamples. We shall present moreover global Lipschitz regularity results and boundary estimates in the context of convex domains for $k=1$, leading to the existence of a principal eigenfunction. Some open question will be raised. This is joint work with I. Birindelli and H. Ishii.

Mathematics; Partial differential equations; Integral equations

video/mp4

Author affiliation: Sapienza Università di Roma

2017-10-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63169

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/