Abstract

Inverse Boundary Problems for Acoustic and Heat Equations (Moments' Method) Y. V. Kurylev We consider two inverse problems: (A) Inverse boundary spectral problem for the acoustic operator $$ Au = - c^2(x) \Delta u; \quad u|_{\partial M} = 0 $$ where $M$ is a bounded domain in $R^n, n \geq 2$ with smooth boundary. (Joint results with A. Starkov and K.Peat) (B) Inverse boundary problem for the heat equation $$\kappa u_t - \Delta u = 0; \quad u|_{t=0} = 0, \quad u|_{\partial M \times (0, \infty)} = f.$$ (Joint results with M.Kawashita and H.Soga) The data used are: (A) First $N$ eigenvalues $\lambda_k, k = 1,...,N$ and normal derivatives on $\partial M$ of the corresponding eigenfunctions $\phi_k$ which are assumed to be known with some error; (B) Normal derivatives on $\partial M \times (0,t_0)$ of the solutions to the heat equations which correspond to $f = P_{i,K}|_{\partial M} \times H(t)$. Here $P_{i,K}(x)$ is a basis of harmonic polynomials of order $K$ (or less) and $H(t)$ is the Heaviside function. It is assumed that these normal derivatives are again known with some error. We develop a procedure to approximately reconstruct $c(x)$ or $\kappa(x)$ and analyse its stability (with respect to $N, K, t_0$ and also error estimates for the data) under some assumption upon smoothness of $c(x)$ or $\kappa(x)$.

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