scholarly journals Uniform bounds for solutions to elliptic problems on simply connected planar domains

2019 ◽  
Vol 147 (10) ◽  
pp. 4289-4299 ◽  
Author(s):  
Luca Battaglia
Keyword(s):  
Elliptic Problems ◽  
Uniform Bounds ◽  
Planar Domains ◽  
Simply Connected

scholarly journals On the distribution of harmonic measure on simply connected planar domains

2003 ◽  
Vol 75 (2) ◽  
pp. 145-152 ◽  
Author(s):  
Dimitrios Betsakos ◽  
Alexander Yu. Solynin
Keyword(s):  
Harmonic Measure ◽  
Planar Domain ◽  
Planar Domains ◽  
Simply Connected

AbstractFor a simply connected planar domain D with 0 ∈ D and dist(0, ∂D) = 1, let hD(r) be the harmonic measure of ∂ D ∩{|Z| ≤ r} evaluated at 0. The function hD(r) is the distribution of harmonic measure. It has been studied by B. L. Walden and L. A. Ward. We continue their study and answer some questions raised by them by constructing domains with pre-specified distribution.

From Steklov to Neumann and Beyond, via Robin: The Szegő Way

2019 ◽  
Vol 72 (4) ◽  
pp. 1024-1043 ◽  
Author(s):  
Pedro Freitas ◽  
Richard S. Laugesen
Keyword(s):  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Steklov Eigenvalue ◽  
Robin Laplacian ◽  
Fixed Area ◽  
Neumann Laplacian ◽  
Simply Connected ◽  
Second Eigenvalue ◽  
Simply Connected Domains

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

scholarly journals Principal frequency of the p-Laplacian and the inradius of Euclidean domains

2015 ◽  
Vol 07 (03) ◽  
pp. 505-511 ◽  
Author(s):  
Guillaume Poliquin
Keyword(s):  
Lower Bound ◽  
Laplace Operator ◽  
Lower Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Principal Frequency ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Euclidean Domains

We study the lower bounds for the principal frequency of the p-Laplacian on N-dimensional Euclidean domains. For p > N, we obtain a lower bound for the first eigenvalue of the p-Laplacian in terms of its inradius, without any assumptions on the topology of the domain. Moreover, we show that a similar lower bound can be obtained if p > N - 1 assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains.

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