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

Abstract

We consider the ill-posed Cauchy problem in a bounded domain D of Rn for an elliptic differential operator A(x,∂) with data on a relatively open subset S of the boundary ∂D. We do it in weighted Sobolev spaces Hs,γ(D) containing the elements with prescribed smoothness s∈N and growth near ∂S in D, controlled by a real number γ. More precisely, using proper (left) fundamental solutions of A(x,∂), we obtain a Green-type integral formula for functions from Hs,γ(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 Hs,γ(D) whenever this solution exists.