scholarly journals Remarks on the metric induced by the Robin function

2011 ◽  
Vol 60 (3) ◽  
pp. 751-802
Author(s):  
Diganta Borah ◽  
Kaushal Verma
Keyword(s):  
Robin Function

scholarly journals The Robin Function and Its Eigenvalues

2007 ◽  
Vol 14 (3) ◽  
pp. 403-417
Author(s):  
Bodo Dittmar ◽  
Maren Hantke
Keyword(s):  
Eigenvalue Problem ◽  
Connected Domain ◽  
Main Part ◽  
Eigenvalue Problems ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Simply Connected Domain ◽  
Robin Function ◽  
Membrane Problem ◽  
Simply Connected

Abstract The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Green's function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.

The Robin function and conformal welding – A new proof of the existence

2020 ◽  
Vol 27 (1) ◽  
pp. 43-51
Author(s):  
Bodo Dittmar
Keyword(s):  
Conformal Mapping ◽  
Boundary Value Problem ◽  
Harmonic Functions ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Uniformization Theorem ◽  
Robin Function ◽  
Connected Case ◽  
Simply Connected ◽  
Mathematical Physicist

AbstractGreen’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

scholarly journals Non Degeneracy of Critical Points of the Robin Function with Respect to Deformations of the Domain

2013 ◽  
Vol 40 (2) ◽  
pp. 103-116 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Critical Points ◽  
Robin Function

On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

2021 ◽  
Vol 127 (2) ◽  
pp. 337-360
Author(s):  
Norman Levenberg ◽  
Franck Wielonsky
Keyword(s):  
Convex Body ◽  
Unit Ball ◽  
General Formula ◽  
Plurisubharmonic Function ◽  
Integral Formula ◽  
Compact Set ◽  
Transfinite Diameter ◽  
Robin Function ◽  
Euclidean Unit Ball ◽  
Definition Of

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

Modified harmonic Robin function

2011 ◽  
Vol 58 (4) ◽  
pp. 483-496 ◽  
Author(s):  
H. Begehr ◽  
T. Vaitekhovich
Keyword(s):  
Robin Function

PROFILE AND EXISTENCE OF SIGN-CHANGING SOLUTIONS TO AN ELLIPTIC SUBCRITICAL EQUATION

2008 ◽  
Vol 10 (06) ◽  
pp. 1183-1216 ◽  
Author(s):  
MOHAMED BEN AYED ◽  
RABEH GHOUDI
Keyword(s):  
Critical Exponent ◽  
Elliptic Problem ◽  
Blow Up ◽  
Low Energy ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Robin Function ◽  
Sign Changing Solutions ◽  
Two Bubbles

In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pε) : Δ2 u = |u|(8/(n-4))-εu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We characterize the low energy sign-changing solutions (uε) of (Pε). We prove that (uε) are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Furthermore, we construct families of each kind of these solutions and we prove that the bubble-tower solutions exist in our case.

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