This thesis establishes new regularity results for three classes of partial differential equations. First, we prove second order Holder regularity for a semi-linear variant of the bi-Laplacian equation in the superlinear, subquadratic setting. Second, we investigate a dead core problem for a degenerate/singular normalized p-Laplacian with power-type degeneracy. We derive geometric results, and at the boundary of the positivity set we establish regularity estimates, including non-degeneracy, free boundary porosity, and a Strong Maximum Principle for the critical case. Finally, we establish sharp Hölder regularity for the gradient of viscosity solutions to degenerate/singular normalized p-Laplacian equations, where the degeneracy covers a general class, and prove Sobolev estimates for the homogeneous equation. As an application, for p sufficiently close to 2, we use the Sobolev estimates to demonstrate a known conjecture for the p-Laplacian.

Banca:

Orientador: Boyan Sirakov - PUC-Rio

Co-orientador: Makson Sales Santos - Inst. Sup. Téc. - Lisboa

Eduardo V. Oliveira Teixeira - University of Central Florida

Edgard Almeida Pimentel - Universidade de Coimbra

Damião Júnio Gonçalves Araújo - UFPB

José Miguel Urbano - Universidade de Coimbra

José Vitor da Silva - UniCamp