By Yakov Roitberg (auth.)

This monograph provides elliptic, parabolic and hyperbolic boundary worth difficulties for structures of combined orders (Douglis-Nirenberg systems). For those difficulties the `theorem on entire selection of isomorphisms' is confirmed. a number of functions in elasticity and hydrodynamics are taken care of. The publication calls for familiarity with the weather of sensible research, the speculation of partial differential equations, and the idea of generalized capabilities. *Audience:* This paintings might be of curiosity to graduate scholars and study mathematicians focused on parts corresponding to useful research, partial differential equations, operator idea, the math of mechanics, elasticity and viscoelasticity.

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**Example text**

Ph'C~V) = h=l holds. In addition, 1J't+ is finite-dimensional. 0 (\Iv E 1J't+). 22) 32 CHAPTER 1 Proof. 17) is true for u E jjT+&,p,(r) and v E (coo(G))N. Then, the necessity directly follows from obtained Green's formula. Let us prove the sufficiency. 11) hold. 23) 0 h=l (V'V E m*, u E (C OO (G) )N) . 23) with u E (Cgo(G))N directly implies that (u, I+Vo) = 0, Therefore, I+Vo = 0 (V'V E m*). 23) that (7]", MVo) = 0 (V'V E m*). Therefore, (MVo, A ) = O. 16) is r-complete, the range {Bul&G : u E jjT+s,p,(r)} coincides with RAg = Vq,<1,p (V'q ER).

The condition (Mv,IJ1: A ) = ois equivalent to the relation i = 1, .. ,q. Let L = L( e~, ... , e~) denote a linear hull spanned on the elements e~, ... , e~. Since L is finite-dimensional, every element v E HT',2( G) can be represented in the form of the sum v = v' + v", v' E L, (v", L) = O. 35) is already established for the element v". It follows from the finite dimensionality of L that it is true also for v = v' +v" E HT',2(G). This copletes the proof of the lemma. 4. 2) be elliptic, and let 1J1:A+ O.

5. Under the condition 0/ the mapping u ....... p>O = 0, b'vl8G = ° k + 1 - l/p < 0, } H6-S,p. 25) was established in [BKR] (see also [Ber, Ch. III, §6, Subsec. 5». 8. Let us show an example illustrated the using of the graph method. 24). Let Y"p be a functional Banach space such that (COO(G»N c y. ~ 6- S 3 ,p/M6~' and let the space (coo(G»N is densein Y. For example, y. +f,p , N 6 3 f t> r=1 ° . ,p,(r). ',P x rr B h=1 6 - U ",-1/ P,P(öG) . 26) was obtained by Lions and Magenes for S - SN < 0 and Y"p = Lp(G) (see, for details, [R12], [R1]).