Behavior near the boundary of positive solutions of second order parabolic equations. II

Abstract

Abstract. A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder Q = Ω × (0, T ) which vanish on ∂xQ = ∂Ω × (0, T ), where Ω is a bounded Lipschitz domain in R n . This inequality is applied to the proof of the Hölder continuity of the quotient of two positive solutions vanishing on a portion of ∂xQ.

Positive solutions to elliptic equations in unbounded cylinder

“…From Theorem 3.3 in Fabes et al [5] in the non-divergence case, and Theorem 1.1 in Caffarelli et al [2] in the divergence case, we have the following Harnack principle in the boundary.…”

“…Moreover, Boundary Harnack principle also holds for the positive solutions of equation (1). From Theorem 4.3 in Fabes et al [5] in the non-divergence case, and Theorem 1.4 in Caffarelli et al [2] in the divergence case, we have: Lemma 2.6. There is a universal constant C > 0 which is only dependent of n, Λ,…”

“…The maximum principle for the solutions in cylinder is proved under a bounded condition. We then introduce a so-called boundary Harnack principle which is proved in Fabes et al [5](Theorem 4.3) in the non-divergence form and in Caffarelli et al [2](Theorem 1.4) in the divergence form for the positive solutions of equation (1). Through them we are able to compare the solutions.…”

Upper estimates for inhomogeneous random walks confined to the positive orthant

“…] multiplied by a factor ρ = ρ(d, α, Γ) < 1. An argument from Fabes, Safonov and Yuan [13] allows then to deduce the Carleson inequality.…”

“…where A denotes a domain in Z d and a < b ∈ Z, we define the lateral boundary ∂ l B and the parabolic boundary ∂ p B of B by ∂ More generally, we can define the parabolic boundary of a space-time domainD ⊂ Z d ×Z by ∂ p D = {(x, t) ∈ D c , (x + e, t + 1) ∈ D,for some e ∈ Γ}. This definition is the natural extension of the notion of parabolic boundary of a domain D ⊂ R d × R used in the theory of second order parabolic equations[13], which is defined as the set of all points (x, t) ∈ ∂D such that there is a continuous curve lying in D ∪ {(x, t)} with initial point at (x, t) along which t is non-decreasing. This explains why A × {b} is not included in ∂ p B Let L be a difference operator as in (2.2), let A ⊂ Z d denote a domain in Z d and f a real valued function defined on A.…”