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

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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.

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