Boundary estimates for non-negative solutions to non-linear parabolic equations

Abstract

This paper is mainly devoted to the boundary behavior of non-negative solutions to the equation

$$\begin{aligned} \mathrm{H }u =\partial _tu-\nabla \cdot \mathrm{A }(x,t,\nabla u) = 0 \end{aligned}$$

in domains of the form $\Omega _T=\Omega \times (0,T)$ where $\Omega \subset \mathbb R^n$ is a bounded non-tangentially accessible (NTA) domain and $T>0$. The assumptions we impose on $A$ imply that $H$ is a non-linear parabolic operator with linear growth. Our main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerning the boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type $\partial _tu-\nabla \cdot (\mathrm{A }(x,t)\nabla u)=0$, where $\mathrm{A }$ is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green’s function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.