In this chapter we collect several results concerning direct problems that
are used in the following exposition, especially in Chapters 3, 4, and 5. Since
proofs in most cases are rather difficult and long we give only formulations re-
ferring the reader to accessible and well-known books and articles. However,
where it is inconvenient or impossible to find results or their proofs we give
proofs as well. In summary, we show that direct problems are well posed and
sometimes describe their refined properties. We hope the collection presented
will be useful in a subsequent study of inverse problems.
1.1. Sets and functions. Differential operators. Let R" denote the real n-
dimensional space of points x = (x{, ... , xn) and
denote the complex
space of points z = (z{, ... , zn). We set 'x = (0, x2, ... , xn). If z ,
w G C" we define (z 9w) = zlwlA
define \x\ to be the Euclidean
length (x,
of a vector x
, and let B(x; r) denote the ball {y e
\x - y\ r) centered at x and of radius r. Let E = dB(0; 1).
If S is a set in R" , then dS, 5 or clS\ intS\ and V(S) denote its
boundary, closure, interior, and neighborhood respectively. Q and G stand
for open sets. An open connected set is said to be a domain. The char-
acteristic function /(5) is defined as 1 on S and 0 on
\S. The k-
dimensional Lebesgue measure or surface measure is denoted by measfc and
the ^-dimensional one by dx . We define on measA2_1 X.
We shall use a multi-index a (a{, ... , an) with nonnegative integer
components a and define
Dj = d/OXj ,
= D^
| =
+ + an.
For a function u on a set S and a number A, 0 X 1, we introduce
the usual Holder norm:
\u\x(S) = swp\u(x)-u(y)\/\x-y\ +sup\u(x)\, sup over x, y e S, x^y.
If X = 0 we drop the first term of the right-hand side. For a nonnegative k
we let
\u\k(Q)= J2
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