scholarly journals Eigenvalues of the Laplacian with Neumann boundary conditions

1985 ◽  
Vol 26 (3) ◽  
pp. 293-309 ◽  
Author(s):  
H. P. W. Gottlieb
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Summation Formula ◽  
Laplacian Operator ◽  
Eigenvalues Of The Laplacian ◽  
Annular Sector ◽  
Simply Connected

AbstractVarious grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

scholarly journals Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

2010 ◽  
Vol 62 (4) ◽  
pp. 808-826
Author(s):  
Eveline Legendre
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
First Eigenvalue ◽  
Bounded Number ◽  
Eigenvalues Of The Laplacian ◽  
Neumann Laplacian ◽  
Dirichlet And Neumann Conditions ◽  
Second Eigenvalue

AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

scholarly journals Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions

2019 ◽  
Vol 39 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D'Aguì ◽  
Angela Sciammetta
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Mixed Boundary Conditions ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
Differential Problem ◽  
Nonlinear Elliptic

In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.

scholarly journals Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

2019 ◽  
Vol 150 (1) ◽  
pp. 475-495 ◽  
Author(s):  
Begoña Barrios ◽  
Maria Medina
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Classical Case ◽  
Parabolic Problems ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Local Version ◽  
Comparison Results ◽  
Non Local

AbstractWe present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical cases= 1 in [23, 24] respectively.

Study of Stokes dynamical system in a thin domain with Fourier and Tresca boundary conditions

2019 ◽  
pp. 2150007 ◽  
Author(s):  
Y. Letoufa ◽  
H. Benseridi ◽  
M. Dilmi
Keyword(s):  
Boundary Conditions ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Dynamic Regime ◽  
Thin Domain ◽  
Problem Statement ◽  
Stokes Fluid

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain [Formula: see text] with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

Three-Dimensional Vibration Analysis of Rectangular Plates With Mixed Boundary Conditions

2005 ◽  
Vol 72 (2) ◽  
pp. 227-236 ◽  
Author(s):  
D. Zhou ◽  
Y. K. Cheung ◽  
S. H. Lo ◽  
F. T. K. Au
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Mixed Boundary ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Boundary Function ◽  
Rectangular Plates ◽  
Characteristic Boundary ◽  
Fixed Boundaries

Three-dimensional vibration solutions are presented for rectangular plates with mixed boundary conditions, based on the small strain linear elasticity theory. The analysis is focused on two kinds of rectangular plates, the boundaries of which are partially fixed while the others are free. One of those studied is a rectangular plate with partially fixed boundaries symmetrically arranged around four corners and the other one is a rectangular plate with partially fixed boundaries around one corner only. A global analysis approach is developed. The Ritz method is applied to derive the governing eigenvalue equation by minimizing the energy functional of the plate. The admissible functions for all displacement components are taken as a product of a characteristic boundary function and the triplicate Chebyshev polynomial series defined in the plate domain. The characteristic boundary functions are composed of a product of four components of which each corresponds to one edge of the plate. The R-function method is applied to construct the characteristic boundary function components for the edges with mixed boundary conditions. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The influence of the length of the fixed boundaries and the plate thickness on frequency parameters of square plates has been studied in detail. Some valuable results are given in the form of tables and figures, which can serve as the benchmark for the further research.

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