Study of Solvability of the Dirichlet Problem for Nonlinear Differential Equations of Infinite Order by the Method of Selection Main Part.

This paper studies the solvability of the Dirichlet problem for nonlinear differential equations of infinite order. Previously, this was considered a sequence of truncated equations and boundary conditions of order $2m$ and using the limit transition as the m tends to infinity have established the existence of a generalized solution of the original problem. In this article we propose a new approach, namely:to study the solvability of the Dirichlet problem for nonlinear differential equations of infinite order is proposed to introduce a differential operator of infinite order as a sum of two operators of infinite order, of which one main and the other subordinate to him. The basis of their comparison, the expected ratio of the corresponding energy of a Sobolev space of infinite order. Then when certain conditions for main and subordinate operators are able to prove a theorem on the existence of a generalized solution to the original equation, with any right part from the space conjugate to the space corresponding to the main operator. Namely, the main operator is the operator, the energy space which is compactly embedded in energy space corresponding to the second (called subordinates). Embedding theorems and compact attachment of a Sobolev space of infinite order, like a onedimensional and multidimensional regions, rather fully developed by the author.
The obtained results are illustrated on the example of a particular Dirichlet problem for a differential equation of infinite order on two-dimensional square. It is shown how for a given two-dimensional matrix, describing the main operator and containing infinitely many zeros, built a regularized matrix with positive terms, defining the space of the matching element by element energy space corresponding to the main operator. This allowed just install the subordination of one operator to another and to check that all conditions of the theorem.

Keywords: solvability, embedding theorems, spaces, infinite order, subordinate operator.