By Ivan Fesenko

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**Example text**

7. Then for any f E H t ( G ) the solution of the equation Acu = f belongs t o Hs+t(G)if and only if C [ H s ( r ) ] C q [Hs+t(r)]q. In particular, D(&) c H t ( G ) if and only if R ( C ) G [ H t ( r ) ] q . These results admit generalizations to the case of matrix elliptic differential operators A ( z ,D ) , and the case of L,-spaces instead of L2-spaces. Moreover, conditions for the Fkedholm property and other types of solvability of realizations are also described. In Sect. 6 we will discuss some spectral properties of these realizations.

In particular, C is a triangular matrix. Generally, in distinction t o the case ml = . . = mp,C is not invertible. Now we set and denoting the vector-valued function (u1,.. 42) Po. 47) M. S. Agranovich 62 I. Elliptic Boundary Problems where E , S E {0,1}. The reduced Cauchy data for u relative to A and for v relative t o A* are defined by the formulas where I x are m x m skew-unit block matrices. Here xouis the Dirichlet data of u,and xlu,or rather Ixxlu, is a choice of reduced Neumann data of u.

11), and let w be a function from H q ( M ) satisfying the stable boundary conditions. 10) of the boundary problem. However now the space ‘Ft of solutions is defined as follows: ‘H = { u E Hq(G): B ~ = u . . = B,-lu = 0 on r }. 10). 7) is valid with some positive E and nonnegative C1. The term was introduced by Aronszajn. The conditions for coerciveness have been studied by Aronszajn (Aronszajn 1955) and other authors. 4. The form a[u,v] is coercive on the space 3-1 i f the following two conditions hold.

### Number theory 1998 by Ivan Fesenko

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