scholarly journals Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity

2007 ◽  
Vol 185 ◽  
pp. 143-150 ◽  
Author(s):  
Zbigniew Błocki
Keyword(s):  
Green Function ◽  
Bounded Domain ◽  
Bergman Kernel ◽  
Main Tool ◽  
Logarithmic Capacity ◽  
Bergman Metric ◽  
Similar Estimate ◽  
Several Variables ◽  
The Green Function ◽  
Complex Laplacian

AbstractFor a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

The Bergman kernel and the green function

1997 ◽  
Vol 87 (2) ◽  
pp. 3366-3380
Author(s):  
V. A. Malyshev
Keyword(s):  
Green Function ◽  
Bergman Kernel ◽  
The Green Function

Boundary behavior of Green function for a parabolic equation in bounded domain

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Ghanmi Abdeljabbar ◽  
Tarek Kenzizi
Keyword(s):  
Parabolic Equation ◽  
Green Function ◽  
Bounded Domain ◽  
Boundary Behavior ◽  
The Green Function

AbstractThe purpose of the paper is to describe the boundary behavior of the Green function of the parabolic equation Δ

scholarly journals Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J. 129 (1993), 43–52

1995 ◽  
Vol 137 ◽  
pp. 145-148 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Green Function ◽  
Kernel Function ◽  
Bergman Kernel ◽  
Holomorphic Functions ◽  
Bergman Kernel Function ◽  
Nagoya Math ◽  
Sublevel Sets ◽  
Hyperconvex Domain ◽  
The Green Function

0. In [0-1] it was proved that for any bounded hyperconvex domain D in C2 the Bergman kernel function K(z, w) of D satisfiesIn case n ═ 1, this is due to a behavior of sublevel sets of the Green function. The general case then follows by the extendability of L2 holomorphic functions.

scholarly journals Localization lemmas for the Bergman metric at plurisubharmonic peak points

2003 ◽  
Vol 171 ◽  
pp. 107-125 ◽  
Author(s):  
Gregor Herbort
Keyword(s):  
Green Function ◽  
Positive Constant ◽  
Bergman Kernel ◽  
Extremal Function ◽  
Additional Assumption ◽  
Bergman Metric ◽  
Pluricomplex Green Function ◽  
Mass Property ◽  
Peak Points ◽  
The Extremal Function

AbstractLet D be a bounded pseudoconvex domain in ℂn and ζ ∈ D. By KD and BD we denote the Bergman kernel and metric of D, respectively. Given a ball B = B(ζ, R), we study the behavior of the ratio KD/KD∩B(w) when w ∈ D ∩ B tends towards ζ. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to 1, as w tends to ζ, under an additional assumption on the pluricomplex Green function D(·, w) of D with pole at w, namely that the diameter of the sublevel sets Aw :={z ∈ D | D(z, w) < −1} tends to zero, as w → ζ. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in [DiOh1], where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set Aw is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function
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