An inverse problem originating from magnetohydrodynamics

Abstract

We are concerned with the possibility of identifying the real parameters a and b on the right-hand side of the equation u = au + b 0 inω R 2 (0.1) for a function u satisfying the boundary conditions u = 0, ∂u/∂ν = on ∂ω (0.2) with any fixed sufficiently smooth function ≡ 0.In the case of a smooth curve γ = ∂ω, we provide a sufficient condition, under which the pair (a, b) can be uniquely reconstructed through the specified function . On the basis of this sufficient condition, we show that there are at most finitely many pairs (a, b) if ω is (simply connected and) different from a disc.If ω has a corner we prove that the pair (a, b) is unique and can be calculated explicitly if is known on one side near the corner.

The recovery of a parabolic equation from measurements at a single point

“…Our question is: What can be said about the composition of the body Ω, its geometry, the thermal coefficients a ij (x), and the internal heat sink coefficient q(x)? This is a typical question faced by physicists working on Tokamak with plasma inside a closed chamber, [10]. The range of the coefficients a ij (x)…”

Функционально-Геометрический Метод Решения Задач Со Свободной Границей Для Гармонических Функций

“…Тогда существует (см. [5], [7], а также [9]) не более конечного числа распределений f u , задаваемых аффинными функциями f : u → f (u) = au + b, если только односвязная область ω не круг (для которого, как легко видеть, число распределений континуально). Метод [5] доказательства этого результа-та базируется на достаточном условии единственности.…”