Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet

Details Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet Homotopies of Eigenfunctions and the Spectrum of the Laplacian on the Sierpinski Carpet

2009-08-20

arxiv.org/abs/0908.2942v1

Mathematics

Analysis of PDEs

54 pages, 37 figures, 2 tables, to appear, Fractals

Scientific paper

Consider a family of bounded domains $\Omega_{t}$ in the plane (or more generally any Euclidean space) that depend analytically on the parameter $t$, and consider the ordinary Neumann Laplacian $\Delta_{t}$ on each of them. Then we can organize all the eigenfunctions into continuous families $u_{t}^{(j)}$ with eigenvalues $\lambda_{t}^{(j)}$ also varying continuously with $t$, although the relative sizes of the eigenvalues will change with $t$ at crossings where $\lambda_{t}^{(j)}=\lambda_{t}^{(k)}$. We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has $\Omega_{0}$ equal to a square and $\Omega_{1}$ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at www.math.cornell.edu/~smh82)

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