Vitali Liskevich (Swansea)

Pointwise and gradient estimates for a class of quasi-linear parabolic equations

Abstract: In this talk we will discuss basic regularity properties of second order quasi-linear elliptic and parabolic equation of divergence type. First we present pointwise estimates for solutions of evolutional p-Laplace equations and porous media equations with measure as a forcing term. Next, for the general structure of the quasi-linear elliptic and parabolic equations with lower order terms we establish optimal conditions for local boundedness of solutions, continuity and the validity of the Harnack inequality. Finally, assuming the divergence structure of the main part of the p-Laplace type evolution equation is differentiable, we will present the estimates for the gradients of solutions and establish sufficient conditions for Lipschitz continuity.