Influence of lower-order derivatives on solvability Dirichlet problem for elliptic systems

Abstract

The article considers the Dirichlet problem for a multidimensional elliptic system containing lower-order derivatives. At the present time elliptic systems of second-order partial differential equations with two independent are well known, what cannot be said about multidimensional elliptic systems. The problem of homotopy classification of such systems has not yet been solved. For elliptic systems phenomena are observed in the nature of the solvability of classical boundary systems, which do not occur in the case of one equation. Among such phenomena, the influence of the loss of smoothness and the influence of lower-order derivatives on the solvability of boundary value problems should be noted. The paper proposes a solution to the Dirichlet problem for a multidimensional elliptic system with lower-order terms of a special form in a half-space. Applying the Fourier transform, the solvability problem is reduced to study of a second-order partial differential equation containing lower-order derivatives. Violations of the violation of the Noetherian property of the Dirichlet problem are considered. The result is formulated as a theorem.