By I. Titeux, Yakov Yakubov

PREFACE the idea of differential-operator equations has been defined in a number of monographs, however the preliminary actual challenge which ends up 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. specifically, we learn difficulties that are received from asymptotic growth with scales. right here the equipment of operator pencils and differential-operator equations are used. This e-book is meant for scientists and graduate scholars in practical Analy sis, Differential Equations, Equations of Mathematical Physics, and comparable themes. it'll unquestionably be very precious for mechanics and theoretical physicists. we want to thank Professors S. Yakubov and S. Kamin for helpfull dis cussions of a few elements of the publication. The paintings at the booklet used to be additionally in part supported through the ecu neighborhood software RTN-HPRN-CT-2002-00274. xiii advent In first sections of the advent, a classical mathematical challenge may be uncovered: the Laplace challenge. The area of definition could be, at the first time, an unlimited strip and at the moment time, a quarter. to unravel this challenge, a well-known separation of variables technique should be used. during this method, the constitution of the answer may be explicitly came across. For extra information about the separation of variables approach uncovered during this half, the reader can consult with, for instance, the publication through D. Leguillon and E. Sanchez-Palencia [LS].

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**Example text**

2/14] . 8. OPERATORS IN INTERPOLATION SPACES Let {Eo, Ed and {Fo ,Fd be interpolation couples . By B({Eo,Ed, {Fo,Fd) we denote the set of operators from Eo + E 1 into Fo+ F 1 such that their restrictions on Ek' k = 0,1 , continuously map E k into F k. 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!!

2]). 6. INTERMEDIATE DERIVATIVES OF SMOOTH VECTOR-VALUED FUNCTIONS Let {Eo, Ed be an interpolation couple. Further, let £ = 1,2, ... , and 1 :S p :S 00. Then one sets W;((O , 1); Eo, E 1 ) : = { u(t) I u(t) is an (Eo + Ed - valued function in (0,1) such that u(t) E Lp((O, 1); Eo), u(l)(t) E Lp((O, 1); Ed, IlullwJ((O,l) ;EO,El) := Ilu(t)IILp((O ,l);Eo) + Ilu(£)(t)IILp((O,l) ;E,j} . It is known that WJ((O, 1); Eo , E 1 ) is a Banach space (see, for example, H. 1]). In J. L. Lions and J. Peetre [LiP] and H.

THE STEADY THERMAL CONDUC TION PROBLEM IN A PLAT E 45 The thermal flux can then also be expressed in powers of s : a ie 1 -1 +ai°+ ca1 + . . 1. i = 1,2 ,3. uO (x ,y) = uO (x) . 11) Proof. 15) across each interface. 16) n. 16) , the first term and the last term are null . 1), o= 1 1 3 3 8 °8 ° j=l 8 y, 8YJ - 1 2:2:kij~~dY3 2: C i= l = 1 2: ~ 1 3 ( 8 °) 2 -1 i=l CII V y u OIlL 8 y, ( _ l ,l ) ' dY3 46 2. T HERMAL CONDUCT ION IN A HALF-ST RIP AND A SECTOR where V y is t he gradient vector: OV ov OV ) V yV: = ( 0Yl ' OY2 ' OY3 .