On the Cauchy problem for operators with injective symbols in the spaces of distributions

Abstract

Abstract Let D be a bounded domain in the n-dimensional Euclidian space (n ≥ 2) having smooth boundary ∂D. We indicate appropriate Sobolev spaces with negative smoothness in D in order to consider the non-homogeneous ill-posed Cauchy problem for an overdetermined operator A with injective symbol. We prove that elements of the indicated Sobolev spaces have traces on the boundary. This easily leads to a weak formulation of the Cauchy problem and to the corresponding uniqueness theorem. We also describe solvability conditions of the problem and construct its exact and approximate solutions. Namely, we obtain the Carleman formula recovering a vector-function u from the indicated negative Sobolev class via its Cauchy data on an open connected set Γ ⊂ ∂D and values of Au on the domain D. Some instructive examples are considered.