On the Cauchy Problem for Elliptic Complexes in Spaces of Distributions

TL;DR

This paper investigates the Cauchy problem for elliptic complexes in Sobolev spaces, providing solvability conditions, solution formulas, and boundary trace characterizations, with applications to regularity and explicit solution constructions.

Contribution

It introduces a framework for solving the non-homogeneous Cauchy problem for elliptic complexes using Sobolev spaces of negative smoothness, including boundary trace descriptions and Carleman formulas.

Findings

Necessary and sufficient solvability conditions derived

Explicit formulas for exact and approximate solutions provided

Carleman formulas constructed for boundary data reconstruction

Abstract

Let D be a bounded domain in n-dimensional Eucledian space with a smooth boundary. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A_i} of first order operators. In particular, we describe traces on the boundary of tangential part t_i (u) and normal part n_i(u) of a (vector)-function u from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue spaces L^2(D) we construct the approximate and exact solutions to the Cauchy problem with maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman's formulae for a (vector-) function u from the Sobolev space H^1(D) by its Cauchy data t_i (u) on a subset S on the boundary of the domain D and the values of A_i u in D modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.