scholarly journals The characteristic function of a harmonic function in a locally Euclidean space

1967 ◽  
Vol 23 (2) ◽  
pp. 291-297
Author(s):  
William Johnson
Keyword(s):  
Harmonic Function ◽  
Euclidean Space ◽  
Characteristic Function

scholarly journals On Fourier transform multipliers in Lp

1972 ◽  
Vol 13 (2) ◽  
pp. 219-223
Author(s):  
G. O. Okikiolu
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Numbers ◽  
Measurable Functions ◽  
Image Position

We denote by R the set of real numbers, and by Rn, n ≧ 2, the Euclidean space of dimension n. Given any subset E of Rn, n ≧ 1, we denote the characteristic function of E by xE, so that XE(x) = 0 if x ∈ E; and XE(X) = 0 if x ∈ Rn/E.The space L(Rn) Lp consists of those measurable functions f on Rn such that is finite. Also, L∞ represents the space of essentially bounded measurable functions with ║f║>0; m({x: |f(x)| > x}) = O}, where m represents the Lebesgue measure on Rn The numbers p and p′ will be connected by l/p+ l/p′= 1.

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.

Mean-value theorems for Riemannian manifolds

1982 ◽  
Vol 92 (3-4) ◽  
pp. 343-364 ◽  
Author(s):  
A. Gray ◽  
T. J. Willmore
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Function ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Integrable Function ◽  
Mean Value ◽  
Geodesic Sphere ◽  
Einstein Spaces ◽  
The Mean ◽  
Image Position

SynopsisLet Mm (r, f) denote the mean-value of a real-valued integrable function f over a geodesic sphere with centre m and radius r in an n-dimensional Riemannian manifold M. We obtain an expansion of Mm (r, f) in powers of r, thereby generalizing Pizzetti's formula valid in euclidean space. From this expansion we prove that the propertyfor every harmonic function near m, characterizes Einstein spaces. We define super-Einstein spaces and prove that they are characterized by the property

scholarly journals An Lp-Comparison, $p\in (1,\infty )$, on the Finite Differences of a Discrete Harmonic Function at the Boundary of a Discrete Box

2020 ◽  
Author(s):  
Tuan Anh Nguyen
Keyword(s):  
Harmonic Function ◽  
Euclidean Space ◽  
Unit Ball ◽  
Finite Differences ◽  
Harmonic Functions ◽  
Normal Component ◽  
Discrete Analogue ◽  
Dimensional Euclidean Space ◽  
Dimensional Lattice ◽  
Discrete Harmonic Functions

AbstractIt is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d ≥ 2, the tangential and normal component of the gradient ∇u on the sphere are comparable by means of the Lp-norms, $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) , up to multiplicative constants that depend only on d,p. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the d-dimensional lattice with multiplicative constants that do not depend on the size of the box.

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

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