Calculus of Variations and Geometric Measure Theory

Q. H. Nguyen

Potential estimates and quasilinear parabolic equations with measure data

created by nguyen on 20 Apr 2021


Accepted Paper

Inserted: 20 apr 2021
Last Updated: 20 apr 2021

Journal: Memoris of the AMS
Pages: 126
Year: 2021


In this memoir, we study the existence and regularity of the quasilinear parabolic equations:

$$ut-div(A(x,t,\nabla u))=B(u,\nabla u)+\mu,$$

in either $\mathbb{R}^{N+1}$ or $\mathbb{R}^N\times(0,\infty)$ or on a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$ where $N\geq 2$. In this paper, we shall assume that the nonlinearity $A$ fulfills standard growth conditions, the function $B$ is a continuous and $\mu$ is a radon measure. Our first task is to establish the existence results with $B(u,\nabla u)=
^{q-1}u$, for $q>1$. We next obtain global weighted-Lorentz, Lorentz-Morrey and Capacitary estimates on gradient of solutions with $B\equiv 0$, under minimal conditions on the boundary of domain and on nonlinearity $A$. Finally, due to these estimates, we solve the existence problems with $B(u,\nabla u)=
\nabla u
^q$ for $q>1$.