On regularization of the Cauchy problem for elliptic systems in weighted Sobolev spaces

We consider the ill-posed Cauchy problem in a bounded domain To of R-n for an elliptic differential operator A(x, partial derivative) with data on a relatively open subset S of the boundary partial derivative D. We do it in weighted Sobolev spaces H-s,H- gamma(1)) containing the elements with prescribed smoothness s is an element of N and growth near partial derivative S in D, controlled by a real number gamma. More precisely, using proper (left) fundamental solutions of A(x, partial derivative), we obtain a Green-type integral formula for functions from H-s,H- gamma(D). Then a Neumann-type series, constructed with the use of iterations of the (bounded) integral operators applied to the data, gives a solution to the Cauchy problem in H-s,H- gamma(D) whenever this solution exists.