scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals On the Existence of Various Bounded Harmonic Functions with given Periods

1973 ◽  
Vol 49 ◽  
pp. 143-148
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Functions ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Consider a pair (R, Γ) of a Riemann surface R and a period Γ. By a period Γ we mean a real-valued function Γ(γ) on one-dimensional cycles {γ} of the Riemann surface R.

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

scholarly journals Some Theorems on Open Riemann Surfaces

1951 ◽  
Vol 3 ◽  
pp. 141-145 ◽  
Author(s):  
Masatsugu Tsuji
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Green’S Function ◽  
Green's Function ◽  
Riemann Surfaces ◽  
Dirichlet Integral ◽  
Open Riemann Surface ◽  
Surface Spread

Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on Fand be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

1970 ◽  
Vol 22 (4) ◽  
pp. 847-854 ◽  
Author(s):  
Y. K. Kwon ◽  
L. Sario
Keyword(s):  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Higher Dimensions ◽  
Classification Theory ◽  
Superharmonic Function ◽  
Maximum Principles ◽  
Topological Properties ◽  
Positive Continuous Function ◽  
Dirichlet Integral ◽  
Bounded Harmonic Functions

Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

scholarly journals Domination of the supremum of a bounded harmonic function by its supremum over a countable subset

1987 ◽  
Vol 30 (3) ◽  
pp. 471-477 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Harmonic Function ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Open Unit ◽  
Countable Subset ◽  
Open Unit Disc ◽  
Image Position ◽  
Bounded Harmonic Functions ◽  
Bounded Harmonic Function

For what sequences {an} of points of the open unit disc D does there exist a constant k such thatfor all bounded harmonic functions f on D?

scholarly journals Regularity of solutions to anisotropic nonlocal equations

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1135-1155 ◽  
Author(s):  
Jamil Chaker
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Harmonic Functions ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Stable Processes ◽  
Symmetric Stable Processes ◽  
One Dimensional ◽  
Nonlocal Equations ◽  
Bounded Harmonic Functions

Abstract We study harmonic functions associated to systems of stochastic differential equations of the form $$dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots +A_{id}(X_{t-})dZ_t^d$$ d X t i = A i 1 ( X t - ) d Z t 1 + ⋯ + A id ( X t - ) d Z t d , $$i\in \{1,\dots ,d\}$$ i ∈ { 1 , ⋯ , d } , where $$Z_t^j$$ Z t j are independent one-dimensional symmetric stable processes of order $$\alpha _j\in (0,2)$$ α j ∈ ( 0 , 2 ) , $$j\in \{1,\dots ,d\}$$ j ∈ { 1 , ⋯ , d } . In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems.

Rough isometry and energy-finite solutions for the Schrödinger operator on Riemannian manifolds

2003 ◽  
Vol 133 (4) ◽  
pp. 855-873 ◽  
Author(s):  
Seok Woo Kim ◽  
Yong Hah Lee
Keyword(s):  
Riemannian Manifolds ◽  
Schrödinger Operator ◽  
Sobolev Inequality ◽  
Harmonic Functions ◽  
Schrodinger Operator ◽  
Dirichlet Integral ◽  
Local Conditions ◽  
Complete Riemannian Manifolds ◽  
Bounded Harmonic Functions ◽  
Rough Isometry

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

scholarly journals AdS2 × S2 × CY2 solutions in Type IIB with 8 supersymmetries

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Yolanda Lozano ◽  
Carlos Nunez ◽  
Anayeli Ramirez
Keyword(s):  
Quantum Mechanics ◽  
Riemann Surface ◽  
Central Charge ◽  
Global Solutions ◽  
Infinite Family ◽  
Dimensional System ◽  
One Dimensional

Abstract We present a new infinite family of Type IIB supergravity solutions preserving eight supercharges. The structure of the space is AdS2 × S2 × CY2 × S1 fibered over an interval. These solutions can be related through double analytical continuations with those recently constructed in [1]. Both types of solutions are however dual to very different superconformal quantum mechanics. We show that our solutions fit locally in the class of AdS2 × S2 × CY2 solutions fibered over a 2d Riemann surface Σ constructed by Chiodaroli, Gutperle and Krym, in the absence of D3 and D7 brane sources. We compare our solutions to the global solutions constructed by Chiodaroli, D’Hoker and Gutperle for Σ an annulus. We also construct a cohomogeneity-two family of solutions using non-Abelian T-duality. Finally, we relate the holographic central charge of our one dimensional system to a combination of electric and magnetic fluxes. We propose an extremisation principle for the central charge from a functional constructed out of the RR fluxes.

scholarly journals A note on n-harmonic majorants

1986 ◽  
Vol 34 (3) ◽  
pp. 461-472
Author(s):  
Hong Oh Kim ◽  
Chang Ock Lee
Keyword(s):  
Harmonic Function ◽  
Complex Plane ◽  
Unit Disc ◽  
Dirichlet Integral ◽  
Harmonic Majorant ◽  
Harmonic Majorants

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

scholarly journals Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
M. T. Mustafa
Keyword(s):  
Harmonic Function ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Horizontal Component ◽  
Harmonic Morphisms

For Riemannian manifoldsMandN, admitting a submersionϕwith compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians onMandN, we determine conditions under which a harmonic function onU=ϕ−1(V)⊂Mprojects down, via its horizontal component, to a harmonic function onV⊂N.

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