scholarly journals The Frankel property for self-shrinkers from the viewpoint of elliptic PDEs

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Debora Impera ◽  
Stefano Pigola ◽  
Michele Rimoldi
Keyword(s):  
Harmonic Function ◽  
Euclidean Space ◽  
Half Space ◽  
Direct Proof ◽  
Elliptic Pdes ◽  
Finite Point ◽  
Space Property

AbstractWe show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented.

scholarly journals RETRACTED ARTICLE: A note on the boundary behavior for a modified Green function in the upper-half space

2015 ◽  
Vol 2015 (1) ◽  
Author(s):  
Yulian Zhang ◽  
Valery Piskarev
Keyword(s):  
Green Function ◽  
Euclidean Space ◽  
Half Space ◽  
Boundary Behavior ◽  
Indian Acad ◽  
Boundary Property ◽  
Dimensional Euclidean Space ◽  
Retracted Article ◽  
Green Potential

Abstract Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential.

Minimum growth of harmonic functions and thinness of a set

1984 ◽  
Vol 95 (1) ◽  
pp. 123-133 ◽  
Author(s):  
Jang-Mei G. Wu
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Integral Condition ◽  
Finite Point ◽  
Image Position ◽  
Increasing Function

In [3], Barth, Brannan and Hayman proved that if u(z) is any non-constant harmonic function in ℝ2, ø(r) is a positive increasing function of r for r ≥ 1 andthen there exists a path going from a finite point to ∞, such that u(z) > ø(|z|) on Γ. Moreover, they showed by example that the integral condition above cannot be relaxed.

Almost Continuity of Mappings

1968 ◽  
Vol 11 (3) ◽  
pp. 453-455 ◽  
Author(s):  
Shwu-Yeng T. Lin
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Direct Proof ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Natural Context ◽  
Set Of Points

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].

scholarly journals The number of partitions of a set of N points in k dimensions induced by hyperplanes

1967 ◽  
Vol 15 (4) ◽  
pp. 285-289 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
General Position ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Unordered Pair

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

Isometric immersions into manifolds without conjugate points

1982 ◽  
Vol 92 (2) ◽  
pp. 243-250
Author(s):  
J. Bolton
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Height Function ◽  
Unit Vector ◽  
Conjugate Points ◽  
Isometric Immersions

1. Introduction and statement of results. Let f: Mn-1 → ℝn be an immersion into Euclidean space ℝn. Each unit vector v to ℝn determines a height function bv: ℝn → ℝ. The corresponding half-space Lv = b-1([0, ∞) has boundary Hv = (b−1). and L is a (globally) supporting half-space for M at m є M if (m) є H and f(M) ∩ Lv = f(M) ∩ Hv.

A Comparison Between thin Sets and Generalized Azarin Sets

1975 ◽  
Vol 18 (3) ◽  
pp. 335-346 ◽  
Author(s):  
M. Essén ◽  
H. L. Jackson
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Martin Boundary ◽  
Dimensional Euclidean Space ◽  
Thin Sets

Let Rp(p≥2) denote p-dimensional Euclidean space, D the half space defined by {P = (x1, x2, …, xp) ∊ Rp: xp > 0} and ∂D the frontier of D in Rp. The Martin boundary (see [2]) of D can be identified with ∂D∪{∞}.

scholarly journals Modified Poisson Integral and Green Potential on a Half-Space

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Lei Qiao
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Poisson Integral ◽  
Dimensional Euclidean Space ◽  
Superharmonic Functions ◽  
Growth Properties ◽  
Green Potential

We discuss the behavior at infinity of modified Poisson integral and Green potential on a half-space of then-dimensional Euclidean space, which generalizes the growth properties of analytic functions, harmonic functions and superharmonic functions.

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