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absolute continuity analytic function Banach space bi-Lipschitz transformation boundary values Cauchy sequence Cauchy-Kowalewski theorem completely continuous condition on G converges strongly converges weakly Definition 2.5 differentiability domain G elements elliptic equation employ a partition equation 3.1 equivalent to f everywhere f belongs f in Lp(G flh(x function defined function f functions of class G and satisfy G is Lipschitzian Hilbert space inequality inner product integral interior to G Let f lim f linear transformation Lipschitz domain Lipschitzian domain Lp(R neighborhood norm notation open set partition of unity Poincare's lemma Pp(R proof of Theorem prove Quadratic Forms reduce the theorem result follows satisfies a uniform sequence fn strongly in L2 strongly to f subset Suppose Theorem 2.7 Tu,v u2dx uniform Lipschitz condition unique solution unit sphere vanishes identically variables VPu(x weak convergence weakly to f