Positive Harmonic Functions that Vanish on a Subset of a Cylindrical Surface

Abstract

This paper investigates positive harmonic functions on domains that are complementary to a subset of a cylindrical surface. It characterizes, both in terms of harmonic measure and of a Wiener-type criterion, those domains that admit minimal harmonic functions with exponential growth. Illustrative examples are provided. Two applications are also given. The first of these concerns minimal harmonic functions associated with an irregular boundary point, and amplifies a recent construction of Gardiner and Hansen. The second concerns the possible non-approximability of positive harmonic functions by integrable positive harmonic functions.

Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders

“…Therefore, we only need to prove the right inequality of (6.10). By Remark 2, for x 0 = (x 0 , y 0 ) ∈ C, we have 1 C 11) where C is a universal constant. We suppose by contradiction that the right inequality of (6.10) is not true.…”

“…They showed that there are two special solutions with exponential growth at one end while exponential decay at the other and all the positive solutions are linear combinations of these two. Related work, based on sectors, cones, cylinders or second order elliptic equation with lower order terms and with inhomogeneous term, can be found in [1,10,11,18].…”

Positive harmonic functions on comb-like domains

“…(We recall that a positive harmonic function h on a domain Ω is called minimal if any non-negative harmonic minorant of h on Ω is proportional to h.) Benedicks' criterion is also equivalent to the existence of a harmonic function u on Ω vanishing on ∂Ω and satisfying u(x) ≥ |x N | on Ω, and thus describes when a Denjoy domain behaves like the union of two half-spaces from the point of view of potential theory. Related results, based on sectors, cones or cylinders, may be found in [13,22,19]. The purpose of this paper is to describe what happens in the case of another relative of the infinite cylinder.…”

“…We will prove Theorem 1.1 by combining methods from [16], [13] and [19] with some new ideas. It is known (see [8,10,15]) that the behaviour of minimal harmonic functions on simply connected domains is intimately related to the classical angular derivative problem.…”

Positive solutions to elliptic equations in unbounded cylinder

“…Benedicks' criterion describes when a Denjoy domain behaves like the union of two half-spaces from the point of view of potential theory. Related work, based on sectors, cones or cylinders, may be found in [3,12,7,15]. Landis and Nadirashvili [11] showed that a positive solution to a uniformly elliptic equation in a cone of R n which vanishes at the boundary is unique up to a constant multiple.…”