The local boundedness of gradients of weak solutions to elliptic and parabolic phi-Laplacian systems

TL;DR

This thesis presents a unified method to prove the boundedness of gradients in solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems, using energy inequalities and Di Giorgi's method.

Contribution

It introduces a flexible energy inequality and applies Di Giorgi's method to establish gradient boundedness for a broad class of phi-Laplacian systems.

Findings

Proven boundedness of gradients for weak solutions.

Established a Cacciopoli-type energy inequality with a freely chosen function.

Applied Di Giorgi's method to derive L-infinity estimates.

Abstract

In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional function f which can be chosen freely is proven. Then, Di Giorgi's method is applied using level sets which will lead to L-infinity-estimates on the gradient of the weak solution.