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By Victor Isakov

The mathematical works of S.L. Sobolev have been strongly stimulated by means of specific difficulties coming from functions. The technique and concepts of his well-known booklet ''Applications of practical research in Mathematical Physics'' of 1950 grew to become out to be very influential and are time-honored within the learn of varied difficulties of mathematical physics. the subjects of this quantity situation mathematical difficulties, in general from regulate concept and inverse difficulties, describing numerous tactics in physics and mechanics, particularly, the stochastic Ginzburg–Landau version with white noise simulating the phenomenon of superconductivity in fabrics below low temperatures, spectral asymptotics for the magnetic Schrodinger operator, the idea of boundary controllability for types of Kirchhoff plate and the Euler–Bernoulli plate with quite a few bodily significant boundary controls, asymptotics for boundary price difficulties in perforated domain names and our bodies with diverse sort defects, the Finsler metric in reference to the research of wave propagation, the electrical impedance tomography challenge, the dynamical Lam´e approach with residual pressure, and so on.

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A linear relation u u∗ determining A(Ω) is the Cauchy–Riemann condition du∗ = du. • A(Ω) is a generic algebra: ΩA(Ω) = Ω, GA(Ω) ≡ A(Ω) . • By the maximum principle for analytic functions, Shilov’s boundary coincides with the topological boundary: ∂A(Ω) = ∂Ω = Γ . Hence the algebra A(Γ ) := trA(Ω) = {w|Γ | w ∈ A(Ω)} is isometrically isomorphic to A(Ω). As a result, if Γ and the relation h determining A(Γ ) are given, then one can recover Ω (up to a homeomorphism) and A(Ω) (up to an isometric isomorphism) by the scheme (15) in Subsect.

Commutativity. Pσξ Pσξ = Pσξ Pσξ for all projections in P. 3. Cyclicity. P possesses a cyclic element in H. 4. Exhausting property. PσT = I for all σ. Property 1 is principal: the continuously extending reachable sets correspond to the intuitive image of the waves propagating with finite speed. 3 can be applied to. Property 4 is rather technical and is accepted just for the sake of simplicity: it describes the case where the waves propagate into a bounded domain and exhaust the inner space for sufficiently large times.

Sobolev took part in the numerical solution of huge problems of mathematical physics. From that time on to the end of his life, he had an invariable interest in the discrete approximation of continuum objects, especially in cubature formulas. In the present paper, discrete approximations are not only used, they play a crucial role in obtaining the main results. This paper is devoted to the mathematical study of a boundary value problem for the stochastic Ginzburg–Landau model of superconductivity; we hope it will promote a better understanding of the transitions that occur between the superconducting and nonsuperconducting states.

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