Fixed Membrane Problem Research Articles

Isoperimetric bounds for the first eigenvalue of the Laplacian

In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space Rn with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an embedded closed hypersurface in Rn.

Spatial decay estimates for solutions describing harmonic vibrations in the theory of swelling porous elastic soils

This paper is concerned with the study of the amplitude of the steady-state vibration in a right finite cylinder made of a mixture consisting of three components: an elastic solid, a viscous fluid and a gas. An exponential decay estimate of Saint-Venant type in terms of the distance from one end of the cylinder is obtained from a first-order differential inequality for a cross-sectional area measure associated with the amplitude of the steady-state vibration. The decay constant depends on the excitation frequency, the constitutive coefficients and the first positive eigenvalue for the fixed membrane problem for the cross-sectional geometry. The paper also indicates how to extend the results to a semi-infinite cylinder.

Optimal Sub- or Supersolutions in Reaction-Diffusion Problems

The type of problem under consideration isut=Δu+f(u)inΩ×(0, T)∂u∂n+g(u)=0on∂Ω×(0, T) (*)u(x, 0)=u0(x). Here Ω is a finite domain of RN. The solution of (*) is compared with a corresponding solution of the N-ball or a finite interval whose size depends on different quantities of an associated linear elliptic problem for Ω, such as e.g. the fixed membrane problem. Possible applications include estimates for the blow-up or finite vanishing time.

Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains

We describe the boundary behavior of the nodal lines of eigenfunctions of the fixed membrane problem in convex, possibly nonsmooth, domains. This result is applied to the proof of Payne’s conjecture on the nodal line of second eigenfunctions [P1], by removing theC ∞ smoothness assumption which is present in the original proof of Melas [M].

Extension of an inequality of P�lya-Szeg� to wedge-like domains

In this paper an isoperimetric upper bound of Polya-Szego for the first eigenvalue in the fixed membrane problem is extended to domains contained in wedge.

On the nodal lines of second eigenfunctions of the fixed membrane problem

A well-known conjecture about the second eigenfunction of a bounded domain in ℝ2 states that the nodal line has to intersect the boundary in exactly two points. We give sufficient conditions on the domain for this assertion to hold. For special doubly symmetric domains we also prove that λ2 is simple and that the nodal line of the second eigenfunction lies on one of the axes.

Lower bounds of Cheeger-Osserman type for the first eigenvalue of then-dimensional fixed membrane problem

In this note we derive new lower bounds for the first eigenvalue of the Laplacian in a boundedn-dimensional domain with Dirichlet boundary conditions. The lower bounds obtained are related to those of Cheeger (1970) [2] and Osserman (1977) [6], and they turn out to be sharper when the domain is not too far from being a ball.

Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity

AbstractLower bounds for the eigenvalues of some elliptic equations and elliptic systems over bounded regions are obtained. The bounds are universal in that they depend only upon the volume of the region. Specific applications include the clamped plate, the buckling problem for the clamped plate and the equations of linear elasticity.Our results are consequences of extensions of the methods of Li and Yau (Comm. Mat. Phys. 88 (1983) 309–318) who obtained such results for the eigenvalues of the fixed membrane problem.

On two conjectures in the fixed membrane eigenvalue problem

In this paper we give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:

On the mean value of the fundamental mode in the fixed membrane problem

On the mean value of the fundamental mode in the fixed membrane problem

Some isoperimetric norm bounds for solutions of the helmholtz equation

Previous results of the authors [2] relating theL 1 andL 2 integrals of the first eigenfunction in the two dimensional fixed membrane problem are here extended to the analogous problem in higher dimensions.

An isoperimetric inequality for the first eigenfunction in the fixed membrane problem

An isoperimetric inequality is obtained which relates theL 1-andL 2-integrals of the first eigenfunction in the problem of the vibrating clamped membrane.

Finite Difference Approximations for Eigenvalues of Uniformly Elliptic Operators

where A is the Laplacian and Q the region enclosed by the edge of the membrane. The fixed membrane problem and the analogous problem where A is replaced by a more general uniformly elliptic operator L are the problems to be treated. We approximate the eigenvalues A and eigenfunctions of such problems by the method of finite differences. A uniform mesh is placed on R and at the mesh points L is approximated by a difference operator. This leads to an algebraic eigenvalue problem which is generally easier to solve than the original problem. An important question is then how well the algebraic eigenvalues and eigenvectors approximate the A and their associated eigenfunctions. In this paper we give asymptotic estimates for this error as the mesh width tends to zero. The main results are Theorems 5.1 and 6.4, which are applied to specific examples in ? 7. As early as 1917 Richardson [31] gave an analysis for a difference approximation to the membrane problem on a rectangle (although claiming more general regions could be treated). Richardson showed boundedness and equicontinuity of the discrete eigenvectors with respect to the mesh, proving convergence to a continuous eigenfunction for some subsequence. In 1928, Courant, Friedrichs and Lewy [11], using some of Richardson's ideas, gave a more sophisticated analysis of a difference method for the Dirichlet problem for Laplace's equation on a general region with boundary having piecewise continuously-turning tangent. The boundary values were treated by transferring them to nearby mesh points. It was also indicated how the eigenvalue problem could be treated by the same method. Efimenko [14] in 1938 carried out these indications in detail. Lyusternik [26] has given a complete analysis for the type of approximation which transfers boundary data on regions whose boundaries are sufficiently smooth that any functions continuous on them can be continuously extended to the whole plane. None of the above papers estimate the asymptotic rate of convergence. Weinberger [40], [41] has demonstrated how boundary-transfer type approximations can be used to give upper and lower bounds to the eigenvalues of the membrane and more general problems by a very ingenious use of variational