The De Giorgi method : applications to degenerate PDE

The De Giorgi method was developed in 1957 for showing continuity of non-linear elliptic problems. In this work we will apply generalizations of that method to a variety of degenerate problems. Such problems include first-order equations with negative viscosity, hypoelliptic equations including the nonlocal Focker-Planck equation, and transport-diffusion equations with boundary, for which the diffusion is of critical order and degenerates near the boundary. We will also consider a separate problem in which energy techniques can be brought to bear on a hyperbolic problem, namely the stability of shocks to conservation laws.