By I. Titeux, Yakov Yakubov
PREFACE the speculation of differential-operator equations has been defined in numerous monographs, however the preliminary actual challenge which results in those equations is frequently hidden. whilst the actual challenge is studied, the mathematical proofs are both no longer given or are quick defined. during this e-book, we supply a scientific remedy of the partial differential equations which come up in elastostatic difficulties. particularly, we research difficulties that are bought from asymptotic enlargement with scales. right here the tools of operator pencils and differential-operator equations are used. This booklet is meant for scientists and graduate scholars in useful Analy sis, Differential Equations, Equations of Mathematical Physics, and comparable issues. it'll certainly be very worthwhile for mechanics and theoretical physicists. we wish to thank Professors S. Yakubov and S. Kamin for helpfull dis cussions of a few elements of the e-book. The paintings at the publication used to be additionally partly supported by means of the ecu group software RTN-HPRN-CT-2002-00274. xiii creation In first sections of the creation, a classical mathematical challenge should be uncovered: the Laplace challenge. The area of definition could be, at the first time, an enormous strip and at the moment time, a region. to resolve this challenge, a well-known separation of variables process may be used. during this means, the constitution of the answer will be explicitly discovered. For extra information about the separation of variables technique uncovered during this half, the reader can consult with, for instance, the ebook via D. Leguillon and E. Sanchez-Palencia [LS].
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Extra resources for Application of Abstract Differential Equations to Some Mechanical Problems
T. 29]. 4. Semigroup of linear bounded operators in a Banach space T he t heory of t he semigroup of linear bounded operators in a Ban ach space plays an imp ortant role in many sub jects in mathematics. One can st udy t he th eory in detail in , for exa mple, books by E. Hille and R. S. Phillips [HiP], and J. A. Goldstein [Go]. 1. SEMIGROUP AND GENERATING OPERATOR A famil y of bounded operators U(t ) in a Banach space E depending on the parameter t E (0, 00) is called a semigroup if U(t + T) = U(t)U( T), t > 0, T > 0, and is called a group if t he last equa lity holds for any t E JR, T E lR.
It is known (H. Fo)IITII~(El ,Fl)' 0 < e < 1, 1 ~ p ~ 00. 9. INEQUALITIES Let us state a number of well-known inequalities that are often used : (1) The Young inequality: for 1 < P < 00, ~ + = 1, e > 0, a, b > 0, fJ ab ~ ~(ca)P + 2. (~)pl . c (2) The generalized Holder inequality for functions : In!! N IUk(X)1 dx ~ p' P !! (In N l IUk(X)\Pk dX) Pk , N 1 2:-=1, k=l Pk where n is a bounded domain in ~n , Uk E L Pk(n). (3) For N = 2 and Pk = 2, k = 1,2, the Holder inequality becomes the Cauchy-Schwarz inequality: In lu(x)v(x)1 dx ~ (In 1 lu(xW dX) 2 (In 1 Iv(xW dX) 2 8.
6) 2) for some q E (1,00), operators B k from W;(O, 1) into Lq(O , 1) are compact . 5). Further, for explicitness, we will often write out conditions b) and 2) every tim e they are used, in spite of their being in the definition of p-regularity. 3. ISOMORPHISM AND COERCIVENESS ON T HE WHOLE AXIS 25 Here the case p = 0 or p = n is also admitted. , they are given only in 0 or in 1, th en it follows from the p-regul arity of the boundaryfunctional conditions that the numb er of th em in 0 is equal to p , and in 1 is equal to n - p.