where the kernel [mathematical expression not reproducible].
Using the estimate [mathematical expression not reproducible] then changing [xi] = [eta]y, we obtain
Define the Hilbert space [D.sub.K]([OMEGA]) as the completion of [C.sup.[infinity].sub.c]([OMEGA]) with respect to the norm [mathematical expression not reproducible] induced by the scalar product [mathematical expression not reproducible] given by
where [[absolute value of (u)].sub.p,[OMEGA]] = [([[integral[].sub.[OMEGA]] [[absolute value of (u)].sup.p] dx).sup.1/p], 1 [less than or equal to] p < + [infinity], [D.sub.K]([OMEGA]) [??] [D.sub.k]([OMEGA]) x [D.sub.K]([OMEGA]), endowed with norm [mathematical expression not reproducible], and G(x, s, t) is a nonnegative Caratheodory function from [OMEGA] x R x R to R; namely,
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(respectively, by [mathematical expression not reproducible]);
Then x( m (A)) [mathematical expression not reproducible], and [mathematical expression not reproducible] for each x [member of] A.
X = {[x.sub.1], [x.sub.2],..., [x.sub.j]} : a subset of [mathematical
expression not reproducible], such that [x.sub.i] < [x.sub.h] for all 1 [less than or equal to] i [less than or equal to] h [less than or equal to] j [less than or equal to] n
For any r [member of] N, we define the weighted Sobolev space [H.sup.r.sub.xz] ([LAMBDA]) in the usual way, with the inner product, semi-norm and norm [mathematical
expression not reproducible], respectively.
where [tau] denotes the tangential vector of the edge with [mathematical
expression not reproducible] and (*, *} is the standard Euclidean inner product.
We use the notation [mathematical
expression not reproducible] to denote soft real numbers whereas [bar.x], [bar.y], [bar.z] will denote a particular type of soft real numbers such that [bar.x]([lambda]) = x, for all [lambda] [member of] A etc.
