Zinchenko T. Elliptic systems of differential equations in Hormander spaces

In the theses the properties of general elliptic systems of differential equations are investigated in classes of inner product Hormander spaces over Euclidean space or infinitely smooth closed manifolds. These spaces form the refined Sobolev scale that consists of all Hilbert spaces that are interpolation spaces for pairs of inner product Sobolev spaces. For Douglis-Nirenberg uniformly elliptic systems of differential equations given on Euclidean space, new a priori estimates of solutions in Hormander spaces are established, and the theorems on the solutions regularity are proved. For an extensive class of uniformly elliptic systems on R^n, the theorem on the Fredholm property of the corresponding operator in Hormander spaces is proved. For Douglis-Nirenberg elliptic systems of differential equations given on infinitely smooth closed manifold, the theorems on solvability on the extended Sobolev scale are proved, new a priori estimates of solutions are established, and the theorems on the global and local regularity of solutions on this scale are proved. As an application, new sufficient conditions for the continuity of generalized derivatives of solutions to elliptic systems are found. For broad classes of parameter-elliptic differential equations and systems, the theorems about isomorphisms and a priori estimates of solutions on the extended Sobolev scale are proved.