By Professor Dr. M. M. Lavrentiev (auth.)
This monograph offers with the issues of mathematical physics that are improperly posed within the experience of Hadamard. the 1st half covers numerous ways to the formula of improperly posed difficulties. those methods are illustrated through the instance of the classical improperly posed Cauchy challenge for the Laplace equation. the second one half bargains with a few difficulties of analytic continuations of analytic and harmonic features. The 3rd half is anxious with the research of the so-called inverse difficulties for differential equations within which it truly is required to figure out a dif ferential equation from a undeniable relatives of its suggestions. Novosibirsk June, 1967 M. M. LAVRENTIEV desk of Contents bankruptcy I Formu1ation of a few Improperly Posed difficulties of Mathematic:al Physics § 1 Improperly Posed difficulties in Metric areas. . . . . . . . . § 2 A likelihood method of Improperly Posed difficulties. . . eight bankruptcy II Analytic Continuation § 1 Analytic Continuation of a functionality of 1 advanced Variable from part of the Boundary of the area of Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen § 2 The Cauchy challenge for the Laplace Equation . . . . . . . 18 § three choice of an Analytic functionality from its Values on a suite contained in the area of Regularity. . . . . . . . . . . . . 22 § four Analytic Continuation of a functionality of 2 actual Variables 32 § five Analytic Continuation of Harmonic features from a Circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 § 6 Analytic Continuation of Harmonic functionality with Cylin drical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . forty two bankruptcy III Inverse difficulties for Differential Equations § 1 The Inverse challenge for a Newtonian power . . . . . . .
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Sample text
54) the inequality If(Z)I~exp{-Cl(l-IZI)_ln_e_} InJL;I1(A) is valid, and this concludes the proof. We cite some examples of obtaining fl~(I (A) and the corresponding estimates. 1. Let the set Aw, have positive JORDAN measure fl (Aw,). Then, it is evident that n-+ 00 If, in particular, Aw, consists of m curves, then fl~(! (A) > JL (Awr) JL~11 (A) = , n Let the operator A satisfy the following conditions. 1. For any IPI,1P2,1P2 (x) ~ IPI (x) satisfying a LIPSCHITZ condition with some constant C, the operator A;! Alpl exists where Alp is the FRECHET derivative of the operator A at the point IP. Moreover, for any l/I satisfying a LIPSCHITZ condition with the constant C, the function A;! Alpll/l satisfies a LIPSCHITZ condition with the constant CI depending on C. 2. If 1P2 (x) > IPI (x) the operator A;;Alpl can be represented as A;21Alpll/l = 1 J Qlp2 [x, e, 1P2 (e) -IPI (e)J l/I (~) d~ o where the function Qlp2 (x, ~, 'YJ) is non-negative and continuous for 'Y) > 0. 5. Analytic Continuation of Harmonic Functions from a Circle 39 Theorem 1. 86) 00 L o Let us estimate the function CI~1. 86). 86) the sum to be estimated is bounded. Let the sum reach its conditional maximum over the variables Ck for Ck = Ck (k = 0, ... , (0). 87) hold. Then _ e-r 2q C2 = . r 2p _r 2q ' -2 r 2p -e . 2 P. 90), it follows that ; . -2 L. Cke o 2k ::::; . lne} I . 87) hold. 91) § 5. Analytic Continuation of Harmonic Functions from a Circle 41 from which we obtain ~ C-z 2k {In e ·In Q} L...