Nonnegative solutions of parabolic operators with low-order terms

Lotfi Riahi

doi:10.14232/ejqtde.2003.1.12

Abstract

We develop the harmonic analysis approach for parabolic operator with one order term in the parabolic Kato class on C 1;1 -cylindrical domain . We study the boundary behaviour of nonnegative solutions. Using these results, we prove the integral representation theorem and the existence of nontangential limits on the boundary of for nonnegative solutions. These results extend some rst ones proved for less general parabolic operators.

Highlights

In this paper we are interested in some aspects of the theory of the differential parabolic operator L = ∂ ∂t − div(A(x, t)∇x) +B(x, t).∇x defined on Ω = D×]0, T [, where D is a bounded C1,1-domain of Rn and 0 < T < ∞
In [5], Fabes, Garofalo and Salsa are interested in the same problem for parabolic operators in divergence form with measurable coefficients on Lipschitz cylinders
When they attempted to adapt the techniques of [6] for their case, an interesting difficulty occurs, namely to prove the “doubling” property, which was essential for the proof of Fatou’s theorem and which is equivalent to the existence of a “backward” Harnack inequality for nonnegative solutions

Summary

INTRODUCTION

In [5], Fabes, Garofalo and Salsa are interested in the same problem for parabolic operators in divergence form with measurable coefficients on Lipschitz cylinders When they attempted to adapt the techniques of [6] for their case, an interesting difficulty occurs, namely to prove the “doubling” property, which was essential for the proof of Fatou’s theorem and which is equivalent to the existence of a “backward” Harnack inequality for nonnegative solutions (we refer the reader to [5] for more details). Nystrom studied in [12] parabolic operators in divergence form with measurable coefficients on Lipschitz domains and he proved among other things, the existence and uniqueness of a kernel function and established the integral representation theorem. We deduce a Fatou type theorem for our operator by proving that any nonnegative L-solution on Ω has a nontangential limit at the boundary except for a set of zero L-parabolic measure

NOTATIONS AND KNOWN RESULTS

BOUNDARY BEHAVIOUR

MINIMAL NONNEGATIVE L-SOLUTIONS

INTEGRAL REPRESENTATION AND NONTANGENTIAL LIMITS