Rafael D. Benguria

A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions

*The first authors work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia. He would like to thank the Physics Department and others at the Universidad de Chile for their hospitality during his visit in April, 1990, when much of this research was completed. The second author was supported in part by FONDECYT (Chile), projects 0132-88 and 1238-90. Both authors also thank Fritz Gesztesy for general comments and encouragement.

The Thomas—Fermi—von Weizsäcker Theory of Atoms and Molecules

We place the Thomas-Fermi-von Weizsacker model of atoms on a firm mathematical footing. We prove existence and uniqueness of solutions of the Thomas-Fermi-von Weizsacker equation as well as the fact that they minimize the Thomas-Fermi-von Weizsacker energy functional. Moreover, we prove the existence of binding for two very dissimilar atoms in the frame of this model.

Proof of the Payne-Pólya-Weinberger conjecture

In 1955 and 1956 Payne, Pólya, and Weinberger considered the problem of bounding ratios of eigenvalues for homogeneous membranes of arbitrary shape [PPW1, PPW2]. Among other things, they showed that the ratio k2IK °f * ^ r s t t w o eigenvalues was less than or equal to 3 and went on to conjecture that the optimal upper bound for A2/Aj was its value for the disk, approximately 2.539. It is this conjecture which we establish below. Since 1956 various authors have attempted to prove the conjecture of Payne, Pólya, and Weinberger and some have been able to improve upon the constant 3. Specifically, Brands [Br] in 1964 obtained the value 2.686, de Vries [dV] in 1967 obtained 2.658, and Chiti [Ch2] in 1983 obtained 2.586. In addition, Thompson [Th] gave the natural extension of the PPW argument to dimension n, obtaining

More bounds on eigenvalue ratios for Dirichlet Laplacians in N dimensions

The authors investigate bounds for various combinations of the low eigenvalues of the Laplacian with Dirichlet boundary conditions on a bounded domain

Universal bounds for the low eigenvalues of Neumann Laplacians in N dimensions

\Omega \subset \mathbb{R}^n

On the linear stability theory of Bénard–Marangoni convection

. These investigations continue and expand upon earlier work of Payne, Polya, Weinberger, Brands, Chiti, and the authors of this present paper. In particular, the authors generalize and extend to the n-dimensional setting various bounds of Payne, Polya, Weinberger, Brands, and Chiti and examine their consequences and interrelationships in detail. This includes comparing the asymptotic forms of the various bounds as the dimension n becomes large. The authors also present various extensions and consequences of their recent proof of the Payne–Polya–Weinberger conjecture, including the proof of a second conjecture of Payne, Polya, and Weinberger under an added symmetry condition.

Optimal Bounds for Ratios of Eigenvalues of One-Dimensional Schrodinger Operators with Dirichlet Boundary Conditions and Positive Potentials

The authors consider bounds on the Neumann eigenvalues of the Laplacian on domains in

Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation

I\mathbb{R}^n

A second proof of the Payne-Pólya-Weinberger conjecture

in the light of their recent results on Dirichlet eigenvalues, in particular, their proof of t...

Oscillatory instabilities in the Rayleigh–Bénard problem with a free surface

The linear stability of a fluid bounded above by a free deformable surface is studied numerically. When the heat flux is fixed on the free surface and the lower surface is plane and isothermic, oscillatory instabilities, which may occur at lower values of the Rayleigh number than the critical value for the onset of steady convection, are found.