Alexandra Antoniouk, Nikolai Tarkhanov (auth.), Yuri I.'s Operator Theory, Pseudo-Differential Equations, and PDF

By Alexandra Antoniouk, Nikolai Tarkhanov (auth.), Yuri I. Karlovich, Luigi Rodino, Bernd Silbermann, Ilya M. Spitkovsky (eds.)

This quantity is a set of papers dedicated to the seventieth birthday of Professor Vladimir Rabinovich. the hole article (by Stefan Samko) features a brief biography of Vladimir Rabinovich, besides a few own memories and bibliography of his paintings. it truly is via twenty study and survey papers in numerous branches of research (pseudodifferential operators and partial differential equations, Toeplitz, Hankel, and convolution sort operators, variable Lebesgue areas, etc.) with reference to Professor Rabinovich's examine pursuits. lots of them are written through members of the overseas workshop “Analysis, Operator thought, and Mathematical Physics” (Ixtapa, Mexico, January 23–27, 2012) having a protracted background of medical collaboration with Vladimir Rabinovich, and are in part according to the talks offered there.The quantity should be of serious curiosity to researchers and graduate scholars in differential equations, operator idea, sensible and harmonic research, and mathematical physics.​

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5. 3) in the infinite strip ???? = (−1, 1) × ℝ. We are interested in a solution of this problem in a half-strip ???? ∈ (−∞, ????), where ???? = ????(????(1)). 10 A. Antoniouk and N. Tarkhanov A solution can be found by the Fourier method of separation of variables, see for instance § 2 of Chapter 3 in [28]. We first look for a solution of the corresponding homogeneous problem of the form ????(????, ????) = ????1 (????)????2 (????), obtaining two eigenvalue problems for determining the functions ????1 (????) and ????2 (????). 1) ????1 (±1) = 0.

2) where ???? is a constant and ???? ∈ ???? 1,0 (−∞, ????). Proof. 3. To this end we pick any ???? ∈ ???? 2 (−1, 1). The Sobolev embedding theorem implies that ???? is actually continuous on the interval [−1, 1] and the ????[−1, 1] -norm of ???? older’s is dominated by ???? ∥????∥???? 1 (−1,1) with ???? a constant independent of ????. By H¨ inequality, (∫ 1 )1/2 ∥???? ′ (????)????∥????0 = ∣????′ (????, ????)????(????)∣2 ???????? ′ −1 ≤ ∥???? (⋅, ????)∥????2 (−1,1) ∥????∥????[−1,1], and so ∥???? ′ (????)????∥????0 ≤ ???? ∥????′ (⋅, ????)∥????2 (−1,1) ∥????∥????1 . Hence it follows that ∥???? ′ (????)∥ℒ(????1 ,????0 ) ≤ ???? ∥????′ (⋅, ????)∥????2 (−1,1) , establishing the desired estimate.

N. L. Sobolev and their application to boundary value problems for partial differential equations, Uch. Zapiski Leningr. Ped. Inst. im. I. Gertsena 197 (1958), 54–112. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations in general form, Proc. Steklov Inst. Math. 83 (1965), 3–162. N. A. Samarskii, Equations of Mathematical Physics, 4th Edition, Nauka, Moscow, 1972. [29] N. Wiener, The Dirichlet problem, J. Math. and Phys. Mass. Inst. Tech. 3 (1924), 127–146.

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