Axel Lömker

Bio: Axel Lömker is an academic researcher. The author has contributed to research in topics: Robin boundary condition & Laplace operator. The author has an hindex of 1, co-authored 1 publications receiving 10 citations.

Martin Boundaries of Quasi-Sectorial Domains

Abstract: Cranston and Salisbury have obtained an integral test for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains (cf. [8]). This paper will extend the result as follows: let D ⊂ R d be an open Greenian set (with respect to the Laplacian) consisting of n disjoint open connected cones with Lipschitz boundary and a subset of the boundary of these cones. Let μ be some local Kato measure supported by the boundary of the cones and consider the Schrodinger operator 1/2Δ-µ. We will assume a boundary Harnack principle and give a sufficient integral criterion for the existence of exactly n minimal Martin boundary points at 0. In certain cases there is a necessary criterion, too. When the sufficient integral criterion holds we will give a necessary and a sufficient condition for the existence of a certain process related to the Schrodinger operator that connects two different admissible boundary points. In the paper of Cranston and Salisbury the case μ = 0, d = 2 is treated, but many of the arguments work as well in the general situation.

Martin boundary points of a John domain and unions of convex sets

Abstract: We show that a John domain has finitely many minimal Martin boundary points at each Euclidean boundary point. The number of minimal Martin boundary points is estimated in terms of the John constant. In particular, if the John constant is bigger than $\sqrt3/2$ , then there are at most two minimal Martin boundary points at each Euclidean boundary point. For a class of John domains represented as the union of convex sets we give a sufficient condition for the Martin boundary and the Euclidean boundary to coincide.

Positive harmonic functions of finite order in a Denjoy type domain

Abstract: We introduce a Denjoy type domain and prove that the dimension of the cone of positive harmonic functions of finite order in the domain with vanishing boundary values is one or two, whenever the boundary is included in a certain set.

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.

Positive solutions to elliptic equations in unbounded cylinder

Abstract: This paper investigates the positive solutions for second order linear elliptic equation in unbounded cylinder with zero boundary condition. We prove there exist two special positive solutions with exponential growth at one end while exponential decay at the other, and all the positive solutions are linear combinations of these two.

Martin boundary points of cones generated by spherical John regions

Abstract: Dedicated to Professor Yoshihiro Mizuta on the occasion of his 60th birthday. Abstract. We study Martin boundary points of cones generated by spherical John regions. In particular, we show that such a cone has a unique (minimal) Martin boundary point at the vertex, and also at infinity. We also study a relation between ordinary thinness and minimal thinness, and the boundary behavior of positive superharmonic functions.