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3 Bulletin of the Polish Academy of Sciences. Mathematics

3 2009

3 Ramm A. G.

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Property C for ODE and applications to an inverse problem for a heat equation

100%

Ramm A. G.

2009

tom Vol. 57, no 3-4

243-249

Let lj := -d2 /dx2 + k2qj(x), k = const > 0, j = 1,2, 0 < ess inf qj(x) ≤ ess sup qj(x) < ∞. Suppose that (∗) [integral of, between limits 1 and 0] p(x)u1(x, k)u2(x, k) dx = 0 for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0, 1], which changes sign finitely many times, and uj solves the problem ljuj = 0, 0 ≤ x ≤ 1, u'j(0, k) = 0, uj(0, k) = 1. It is proved that (∗) implies p = 0. This result is applied to an inverse problem for a heat equation.

On the relation between the S-matrix and the spectrum of the interior Laplacian

100%

Ramm A. G.

2009

tom Vol. 57, no 2

181-188

The main results of this paper are: 1) a proof that a necessary condition for 1 to be an eigenvalue of the S-matrix is real analyticity of the boundary of the obstacle, 2) a short proof that if 1 is an eigenvalue of the S-matrix, then k² is an eigenvalue of the Laplacian of the interior problem, and that in this case there exists a solution to the interior Dirichlet problem for the Laplacian, which admits an analytic continuation to the whole space R³ as an entire function.

Attractors of strongly dissipative systems

100%

Ramm A. G.

2009

tom Vol. 57, no 1

25-31

A class of infinite-dimensional dissipative dynamical systems is denned for which there exists a unique equilibrium point, and the rate of convergence to this point of the trajectories of a dynamical system from the above class is exponential. All the trajectories of the system converge to this point as t → +∞, no matter what the initial conditions are. This class consists of strongly dissipative systems. An example of such systems is provided by passive systems in network theory (see, e.g., MR0601947 (83m:45002)).