Isospectral heterogeneous domains: A numerical study

Numerical Solution of Isospectral

In this paper we are concerned with the problem of solving numerically isospectral ows. These ows are characterized by the diierential equation L 0 = B(L);L]; L(0) = L 0 ; where L 0 is a d d symmetric matrix, B(L) is a skew-symmetric matrix function of L and B;L] is the Lie bracket operator. We show that standard Runge{Kutta schemes fail in recovering the main qualitative feature of these ows, ...

Numerical solution of isospectral flows

In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L′ = [B(L), L], L(0) = L0, where L0 is a d × d symmetric matrix, B(L) is a skew-symmetric matrix function of L and [B,L] is the Lie bracket operator. We show that standard Runge–Kutta schemes fail in recovering the main qualitative feature of these...

Numerical solution of isospectral owsMari

Isospectral Domains in Euclidean 3-Space

The question as to whether the shape of a drum can be heard has existed for around fifty years. The simple answer is ‘no’ as shown through the construction of isospectral domains. Isospectral domains are non-isometric domains that display the same spectra of frequencies of sound. These frequencies, deduced from the eigenvalues of the Laplacian, are determined by solving the wave equation in a d...

Compact isospectral sets of plane domains.

Any isospectral family of two-dimensional Euclidean domains is shown to be compact in the C(infinity) topology. Previously Melrose, using heat invariants, was able to establish the C(infinity) compactness of the curvature of the boundary curves. The additional ingredient used in this paper to obtain the compactness of the domains is the behavior of the determinant of the Laplacian near the boun...