Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab

Abstract

This paper investigates the boundary behaviour of positive solutions of the equation L u = 0 Lu = 0 , where L L is a uniformly parabolic second-order differential operator in divergence form having Hölder-continuous coefficients on X = R n × ( 0 , T ) X = {{\mathbf {R}}^n} \times (0,T) , where 0 > T > ∞ 0 > T > \infty . In particular, the notion of semithinness for the potential theory on X X associated with L L is introduced, and the relationships between fine, semifine and parabolic convergence at points of R n × { 0 } {{\mathbf {R}}^n} \times \{ 0 \} are studied. The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of L u = 0 Lu = 0 on X X has parabolic limits Lebesgue-almost-everywhere on R n × { 0 } {{\mathbf {R}}^n} \times \{ 0 \} . Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions.