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Property C for ODE and Applications to an Inverse Problem for a Heat Equation

100%

Ramm A.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

tom 57

nr 3

243-249

Let $ℓ_j:= -d²/dx² + k²q_j(x)$, k = const > 0, j = 1,2, $0 < ess inf q_j(x) ≤ ess sup q_j(x) < ∞$. Suppose that (*) $∫_{0}^{1} p(x)u₁(x,k)u₂(x,k)dx = 0$ for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0,1], which changes sign finitely many times, and $u_j$ solves the problem $ℓ_ju_j = 0$, 0 ≤ x ≤ 1, $u'_j(0,k) = 0$, $u_j(0,k) = 1$. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.

Attractors of Strongly Dissipative Systems

100%

Ramm A.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

tom 57

nr 1

25-31

A class of infinite-dimensional dissipative dynamical systems is defined 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)).

Invisible obstacles

100%

Ramm A.

nr 2

145-148

It is proved that one can choose a control function on an arbitrarilly small open subset of the boundary of an obstacle so that the total radiation from this obstacle for a fixed direction of the incident plane wave and for a fixed wave number will be as small as one wishes. The obstacle is called "invisible" in this case.

A DSM proof of surjectivity of monotone nonlinear mappings

100%

Ramm A.

2009

tom 95

nr 2

135-139

A simple proof is given of a basic surjectivity result for monotone operators. The proof is based on the dynamical systems method (DSM).

Dynamical systems method for solving linear ill-posed problems

100%

Ramm A.

2009

tom 95

nr 3

253-272

Various versions of the Dynamical Systems Method (DSM) are proposed for solving linear ill-posed problems with bounded and unbounded operators. Convergence of the proposed methods is proved. Some new results concerning the discrepancy principle for choosing the regularization parameter are obtained.

A symmetry problem

100%

Ramm A.

nr 1

49-54

Consider the Newtonian potential of a homogeneous bounded body D ⊂ ℝ³ with known constant density and connected complement. If this potential equals c/|x| in a neighborhood of infinity, where c>0 is a constant, then the body is a ball. This known result is now proved by a different simple method. The method can be applied to other problems.

Dynamical systems method for solving linear finite-rank operator equations

63%

Hoang N. , Ramm A.

2009

tom 95

nr 1

77-93

A version of the dynamical systems method (DSM) for solving ill-conditioned linear algebraic systems is studied. An a priori and an a posteriori stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.

Symmetry problems 2

63%

Hoang N. , Ramm A.

2009

tom 96

nr 1

61-64

Some symmetry problems are formulated and solved. New simple proofs are given for some symmetry problems studied earlier. One of the results is as follows: if a single-layer potential of a surface, homeomorphic to a sphere, with a constant charge density, is equal to c/|x| for all sufficiently large |x|, where c > 0 is a constant, then the surface is a sphere.