Regularization of the Cauchy Problem for Elliptic Operators

Abstract

The Cauchy problem for elliptic linear differential operators is a long standing problem connected with numerous applications in physics, electrodynamics, fluid mechanics etc. (see [1,4] or elsewhere). It appears that the regularization methods (see [5]) are most effective for studying the problem. Recently, a new approach was developed, cf. [2] based on the simple observation that the calculus of the solutions to the Cauchy problems foran elliptic equations just amounts to the calculus of a (possibly non-coercive) mixed boundary value problems for an elliptic equations with a parameter. Let D be a bounded domain with Lipschitz boundary ∂D in Euclidean space R, n > 2, with coordinates x = (x1, . . . , xn). For some multi-index α = (α1, . . . , αn) we will write ∂ for