Teoria de Schauder via princípio do máximo, pelo método da compacidade e via potencial newtoniano.

Abstract

The continuity of the Laplacian operator of a function u : Ω ⊂ Rn −→ R does not guarantee that the function is of class C 2 (Ω). However, Schauder’s regularity theory will assure us that if Δu is α-H¨older continuous, then u will be locally C 2,α (Ω), this is, each second-order partial derivative ∂ 2 u ∂xixj , 1 ≤ i, j ≤ n, will be C 0,α loc. In this dissertation, we will establish Schauder’s Theory C 2,α by three methods: by the Maximum Principle, by the Method of Newton’s Compassion and by Potential. In each method, we work on its peculiarities to obtain the desired estimates. Some techniques that we will cover here can be extended to more general operators. On the other hand, all the central difficulties already appear when we consider the Laplacian operator (Poisson equation). For this reason, this is the operator we will consider throughout the work.