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By Paul-Andre Meyer

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0 arbitrarily given. )«||^(G) + 2 ^ | | « | | ^ . 61). 29 that we could work with a finite covering of G. e. 29 depends only on diam(G) and K. 30. 65). 66) holds for all u € W2'P(G) with diam(G) < R. 44) for G = R n and L(x, D) = £ aa{x)Da with l"l=2 Lipschitz continuous coefficients. 31. 65). n) \\u\\w*,r{G) < co(\\L(x, D)u\\LP{G) + \\u\\pLP{G)). 23 with e = \ and we denote it by (tpk)keZn- Further we put Qk := suppV'fc. Note that diam(Qfc) < fn 1 / 2 . pCG) < ^(\\L(x,D)(uipl)\\LP + \\u\\Lp), where \x depends only on p, L(x,D), and n.

Let u € ^ 1 - P ( R " , R ) . Then the function belongs to W 1 ' P (R",R) and we have A((ov«)Al) = R ' *»<«<\ OXJ otherwise [ 0, B. 39) ifu<0 holds. Proof: A. 17, but f(t) = (0 V t) A 1. B. 17. 6 we introduce the concept of a Dirichlet operator in the space L (R",R). Our next aim is to determine classes of second order differential P 30 Chapter 2 Generators of Feller and Sub-Markovian Semigroups operators which might have a closure being a Dirichlet operator. 40) j=l with coefficients a^, bj,cG C&(Rn, R).

1 Second Order Elliptic Differential Operators 33 holds for all x £ Rn and £ 6 Rn. 44) for uniformly elliptic differential operator of second order. Simader. Let us start with some preparatory material. B. 48) and £ ( 4 ( * ) ) 2 = 1. 49) Moreover, there exists N = N(e, n, I) such that the intersection of N + l of the supports of the functions iplk is empty. In fact, we may choose N independent of I. 23. 46). Then we have for all j G No, j < m, fceZ" and £ ii(vi)3*i,P < ^ £ (|)~°~ Jl)p \\urwil,p < oo.

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