Dirichlet Eigenfunctions Research Articles

Pure Point Spectrum for the Laplacian on Unbounded Nested Fractals

We prove for the class of nested fractals introduced by T. Lindstrøm (1990, Memoirs Amer. Math. Soc.420) that the integrated density of states is completely created by the so-called Neuman–Dirichlet eigenvalues. The corresponding eigenfunctions lead to eigenfunctions with compact support on the unbounded set and we prove that for a large class of blow-ups the set of Neuman–Dirichlet eigenfunctions is complete, leading to a pure point spectrum with compactly supported eigenfunctions. This generalizes previous results of H. Teplyaev (1998, J. Funct. Anal.159, 537–567) on the Sierpinski Gasket. Our methods are elementary and use only symmetry arguments via the representations of the symmetry group of the set.

An inverse nodal problem for vectorial Sturm-Liouville equations

Let P(x) denote a 2 × 2 symmetric matrix-valued function defined on [0,1]. We prove that if there exists an infinite sequence {y(x;nj)}j = 1 of Dirichlet eigenfunctions of the operator -(d2)/(dx2)+P(x) whose components all have zeros in common, then P(x) is simultaneously diagonalizable on [0,1]. This result can also be generalized to the general n-dimensional case.

A Harnack inequality for Dirichlet eigenvalues

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 247–257, 2000

Estimates for Dirichlet Eigenfunctions

Estimates for the Dirichlet eigenfunctions near the boundary of an open, bounded set in euclidean space are obtained. It is assumed that the boundary satisfies a uniform capacitary density condition.

Fractal differential equations on the Sierpinski gasket

Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG. We study the analogs of some of the classical partial differential equations with Δ playing the role of the usual Laplacian. For harmonic functions, biharmonic functions, and Dirichlet eigenfunctions of Δ, we give efficient algorithms to compute the solutions exactly, we display the results of implementing these algorithms, and we prove various properties of the solutions that are suggested by the data. Completing the work of Fukushima and Shima [8] who computed the Dirichlet eigenvalues and their multiplicities, we show how to construct a basis (but not orthonormal) for the eigenspaces, so that we have the analog of Fourier sine series on the unit interval. We also show that certain eigenfunctions have the property that they are a nonzero constant along certain lines contained in SG. For the analogs of the heat and wave equation, we give algorithms for approximating the solution, and display the results of implementing these algorithms. We give strong evidence that the analog of finite propagation for the wave equation does not hold because of inconsistent scaling behavior in space and time.

Maxwell and Lamé eigenvalues on polyhedra

In a convex polyhedron, a part of the Lame eigenvalues with hard simple support boundary conditions does not depend on the Lame coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lame coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non-convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non-H2 singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non-convex polyhedron, the spectrum cannot be approximated by finite element methods using H1 elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains. Copyright © 1999 John Wiley & Sons, Ltd.

Dirichlet Eigenfunctions for Horn-Shaped Regions and Laplacians on Cross Sections

Dirichlet Eigenfunctions for Horn-Shaped Regions and Laplacians on Cross Sections

Eigenfunctions of the Koch snowflake domain

We obtain non-tangential boundary estimates for the Dirichlet eigenfunctions ϕn and their gradients {∇ϕn} for a class of planar domains Ω with fractal boundaries. This class includes the quasidiscs and, in particular, snowflake-type domains. When applied to the case when Ω is the Koch snowflake domain, one of our main results states that {∇ϕ1(ω)} tends to ∞ or 0 as ω approaches certain types of boundary points (where ϕ1 > 0 denotes the ground state eigenfunction of the Dirichlet Laplacian on Ω). More precisely, let Ob (resp., Ac) denote the set of boundary points which are vertices of obtuse (resp., acute) angles in an inner polygonal approximation of the snowflake curve ∂Ω. Then given νeOb (resp., νe Ac), we show that {∇ϕ1(ω)}→∞ (resp., 0) as ω tends to ν in Φ within a cone based at ν. Moreover, we show that blowup of {∇ϕ1} also occurs at all boundary points in a Cantor-type set. These results have physical relevance to the damping of waves by fractal coastlines, as pointed out by Sapovalet al. in their experiments on the “Koch drum”.

Sharp estimates for Dirichlet eigenfunctions in Horn-Shaped regions

We prove sharp exponential decay for the Dirichlet eigenfunctions in horn-shaped regions in the plane. The estimate is obtained using a method of Carleman which is widely used in the study of harmonic measure. Such estimates can be applied to study the intrinsic ultracontractivity properties of the heat semigroup for such regions.

Some planar isospectral domains

The authors give a number of examples of two noncongruent isospectral domains in the plane and a particularly simple method of proof. One of their examples is a pair of domains that are not only isospectral but homophonic, i.e. each domain has a distinguished point such that the corresponding normalized Dirichlet eigenfunctions take equal values at the distinguished points. They interprete this to mean that if the corresponding ``drums'' are struck at these special points, then they ``sound the same'' in the very strong sense that every frequency will be excited to the same intensity for each. This shows that one really cannot hear the shape of a drum.

Sharp estimates for Dirichlet eigenfunctions in horn-shaped regions

Sharp estimates for Dirichlet eigenfunctions in horn-shaped regions

The inverse periodic problem for Hill's equation with a finite-gap potential

Let H(q) = d*/d~’ + q(x) be the Hill’s operator with a periodic potential q of period one. Consider the following inverse problem. Find all potentials q(x) with the same periodic spectrum as a given potential p(x). Two explicit solutions are known. For finite-gap potentials, Its and Matveev [6]. give an explicit formula in terms of theta functions. Their solution was generalized by McKean and Trubowitz [lo] to infinite-gap potentials. Finkel et al. [5] give a second formula which expresses q in terms of the base potential p, the Dirichlet eigenfunctions of p and the Floquet solutions of p. The formula involves no algebraic geometry. In this paper we represent finite-gap potentials q as rational functions of the base potential p and the derivatives ofp. The representation makes use of the Finkel-Isaacson-Trubowitz (FIT) formula.

Equidistribution of Neumann data mass on simplices and a simple inverse problem

In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on simplices. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each face is equal to $2/n$ times the $(n-1)$-dimensional volume of the face divided by the volume of the simplex. This is a generalization of \cite{Chr-tri} to higher dimensions. Again it is {\it not} an asymptotic, but an exact formula. The proof is by simple integrations by parts and linear algebra. We also consider the following inverse problem: do the {\it norms} of the Neumann data on a simplex determine a constant coefficient elliptic operator? The answer is yes in dimension 2 and no in higher dimensions.