M511

We introduce the notion of viscosity solutions for second order fully nonlinear elliptic/parabolic equations. A viscosity solution, u, is defined through the condition that whenever a test function, φ, touches u from below/above at a point then φ should be a classical super/sub-solutions of the equation at this point. Since u is apriori merely continuous, it is natural to study whether the equation finally impose any regularizing effects on u. We start by giving an overview of the theory that have been developed the last decades on the interior regularity properties of viscosity solutions, introducing the main tools used, namely the ABP and Harnack inequalities. Then we shall concentrate on oblique derivative problems (where the oblique derivative condition is understood only in the viscosity sense as well) and discuss about the up to the boundary (first order) Hölder regularity of the solutions giving quantitative bounds for the corresponding Hölder norms in terms of the data.

Georgiana Chatzigeorgiou