Regularidade elíptica para modelos não-lineares com condição de bordo oblíquo e aplicações

Abstract

In this Thesis we will study the regularity of viscosity solutions for fully nonlinear elliptic equations with oblique boundary condition. In this regard, firstly, under asymptotic conditions and other hypotheses, Calderon-Zygmund type estimates will be guaranteed for the same viscosity solutions, namely, in context of Lebesgue, weighted Lorentz and weighted Orlicz spaces. The technique used goes back to Tangential Analysis concepts that consist of importing ”fine regularity estimates”of a boundary profile, that is the Recession Operator associated with the second-order original via compactness and stability procedures. Such a process will guarantee such estimates under weakened structural conditions on the operator that governs the problem. Furthermore, we will make some importants applications of this result in a Free Boundary Problem case, in BMO-type Estimates and in density-type theorems of solutions in a general class of viscosity solutions. Finally, we will deal with a study of the optimal regularity of these solutions where the source term will be studied in various integrability scenarios up to the limiting case which would be the case in which such term is in the BMO space.