In this article, we construct aC0 linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a C0 piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the op...

We construct a counterexample to the “hot spots” conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.

We demonstrate that it is possible to compute wave function normalization constants for a class of Schrödinger type equations by an algorithm which scales linearly (in the number of eigenfunction evaluations) with the desired precision P in decimals. Keywords—Eigenvalue problems; Bound states; Trapezoidal rule; Poisson resummation

Spectral properties of some integro-differential operators on R are studied. Characterization of the principal eigenvalue is obtained in terms of the positive eigenfunction. These results are used to prove local and global stability of travelling waves and to find their speed.

The software package SLEDGE for Sturm-Liouville problems attempts to return eigenvalue and eigenfunction estimates within a user requested global error tolerance. The algorithm and underlying mathematics are discussed. Richardson's extrapolation plays a major role in this, but it must be carefully and conservatively implemented.

We investigate stability and approximation properties of the lowest nonzero eigenvalue and corresponding eigenfunction of the Neumann Laplacian on domains satisfying a heat kernel bound condition. The results and proofs in this paper will be used and extended in a sequel paper to obtain stability results for domains in R2 with a snowflake type boundary.

We construct a counterexample to the “hot spots” conjecture; there exists a bounded connected planar domain such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.

In this note we prove a version of the classical Schwarz lemma for the first eigenvalue of the Laplacian with Dirichlet boundary data. A key ingredient in our proof is an isoperimetric inequality for the first eigenfunction, due to Payne and Rayner, which we reinterpret as an isoperimetric inequality for a (singular) conformal metric on a bounded domain in the plane.

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables and eigenfunction expansions in time and space are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.