A boundary harnack principle for lipschitz domains and the principle of positive singularities

Abstract

The primary objective of the present work is the establishment of a "Boundary Harnack Principle" for harmonic functions which vanish on part of the boundary of a Lipschitz domain in Rn and a corresponding result for solutions of the heat equation. The statements of these properties arise naturally from a not-so-familar formulation of the standard Harnack principle, a formulation which is suggested by a necessary condition for the uniqueness statement of the principle of positive singularities.

Boundary behavior and the Martin boundary problem forpharmonic functions in Lipschitz domains

“…To put the results of [LN07] and this paper into perspective, we mention that if p D 2, i.e., in the case of harmonic functions, the term boundary Harnack inequality was first coined in [Kem72] and later proved independently by [Anc78], [Dah77], [Wu78]. This boundary Harnack inequality for positive harmonic functions vanishing on a portion of the boundary of a Lipschitz domain was later extended in [JK82] to nontangentially accessible (NTA) domains, where it is also shown that the corresponding ratio is Hölder continuous.…”

Equivalence between the boundary Harnack principle and the Carleson estimate

“…In fact, when Kemper [Kem72] formulated these notions for the first time, he referred to the global Carleson estimate (Definition 2 below) and the global boundary Harnack principle (Definition 1 below) as the boundary Harnack principle and Property III, respectively. For a Lipschitz domain, Kemper observed that the global Carleson estimate follows from the global boundary Harnack principle and tried to verify the global boundary Harnack principle, though his argument had a gap on [Kem72,page 253]. After Kemper's pioneering work, the global boundary Harnack principle was legitimately proved for a Lipschitz domain by Ancona [Anc78], Dahlberg [Dah77] and Wu [Wu78] independently.…”

Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien

“…Kemper [9] avait eu l'idée d'établir cette propriété pour les fonctions harmoniques ordinaires, mais sa démonstration était incomplète. La méthode que nous utilisons repose sur une ancienne remarque de M. Brelot [1] sur l'action à distance; on étend le contenu de cette remarque aux opérateurs différen-tiels, en montrant que pour les faisceaux harmoniques associés et les faisceaux adjoints il n'y a pas d'action à distance entre deux points frontière distincts; plus précisément, on étend à ces faisceaux une estimation de L. Carleson (voir [3], [10], [11] pour les fonctions harmoniques ordinaires, et [15] pour un faisceau associé à un opérateur elliptique).…”