Regularity theory for nonlinear parabolic PDEs: gradient estimates, stability and the obstacle problem

This thesis concerns different aspects of regularity theory for weak solutions of nonlinear parabolic partial differential equations. We focus on such equations with porous medium type and p-growth structure. In particular, we consider solutions which are defined either via a weak formulation of the equation with test functions under the integral sign, or as functions that obey a parabolic comparison principle. Our research concerns the regularity of both the solution and its gradient. For the gradient of a weak solution of porous medium type systems we prove a higher integrability result up to the boundary of the domain. We derive reverse Hölder inequalities in intrinsic cylinders near the boundary, for which we prove a Vitali type covering property that is applied to obtain the higher integrability result. We also show that under suitable assumptions, weak solutions as well as their gradients are stable with respect to small fluctuations of the parameter characterizing the equation. In particular, we prove that solutions to the approximating problems converge to the corresponding solution of the limit problem in the natural parabolic Sobolev space. For the parabolic p-Laplace equation we study supersolutions, which are defined via a parabolic comparison principle. We show that in the fast diffusion case these functions can be divided into two mutually exclusive classes, for which we give several characterizations. An important tool in regularity theory is the obstacle problem, which is also interesting in its own right. In the case of signed obstacles we study Hölder continuity for solutions to the porous medium type equations defined via a variational inequality. We use a De Giorgi type iteration argument to show that solutions to obstacle problems are locally Hölder continuous, provided that the obstacle is Hölder continuous.