This is the expanded text of a lecture about viscosity solutions of degenerate elliptic equations delivered at the XVI Congresso UMI. The aim of the paper is to review some fundamental results of the theory as developed in the last twenty years and to point out some of its recent developments and applications.
In this paper we are concerned with gradient estimates for viscosity solutions of fully nonlinear second order elliptic equations, generalizing to the nonlinear setting the results of Yanyan Li and Louis Nirenberg about the so-called Glaeser estimate and improving the qualitative results contained in one of our preceding papers.
Con metodi geometrici si stabilisce l'esistenza di soluzioni per sistemi di complementarità degeneri.
We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like where is the mesh step. Such...
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