scholarly journals From Steklov to Neumann via homogenisation

2020 ◽  
Author(s):  
Alexandre Girouard ◽  
Antoine Henrot ◽  
Jean Lagacé
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Eigenvalue Problems ◽  
Perforated Domain ◽  
Intermediate Step ◽  
Planar Domains ◽  
Quantitative Estimates ◽  
Dynamical Boundary Conditions ◽  
Steklov Eigenvalues ◽  
Neumann Eigenvalues

AbstractWe study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.

scholarly journals Integral Harnack inequality

1985 ◽  
Vol 26 (2) ◽  
pp. 115-120 ◽  
Author(s):  
Murali Rao
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Convex Domains ◽  
Dimensional Hausdorff Measure ◽  
Image Position ◽  
Continuous Boundary ◽  
Integral Version

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

A boundary estimate for harmonic functions

1982 ◽  
Vol 91 (1) ◽  
pp. 79-90 ◽  
Author(s):  
P. J. Rippon
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Conformal Mappings ◽  
Dimensional Euclidean Space ◽  
Boundary Behaviour ◽  
Boundary Estimate ◽  
Distortion Theorems

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.

scholarly journals Steklov eigenvalues of reflection-symmetric nearly circular planar domains

2018 ◽  
Vol 474 (2220) ◽  
pp. 20180072 ◽  
Author(s):  
Robert Viator ◽  
Braxton Osting
Keyword(s):  
Asymptotic Estimate ◽  
Perturbation Parameter ◽  
Isoperimetric Inequalities ◽  
Second Order ◽  
Numerical Results ◽  
Domain Perturbation ◽  
Planar Domains ◽  
Constrained Problems ◽  
Area And Perimeter ◽  
Steklov Eigenvalues

We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.

scholarly journals On the derivatives at the origin of entire harmonic functions

1979 ◽  
Vol 20 (2) ◽  
pp. 147-154 ◽  
Author(s):  
D. H. Armitage
Keyword(s):  
Entire Function ◽  
Euclidean Space ◽  
Complex Plane ◽  
Harmonic Functions ◽  
Image Position ◽  
Derivatives Of

If f is an entire function in the complex plane such thatwhere 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

On conjugate harmonic functions in Euclidean space

2002 ◽  
Vol 25 (16-18) ◽  
pp. 1553-1562 ◽  
Author(s):  
F. F. Brackx ◽  
R. Delanghe ◽  
F. C. Sommen
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions
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