Some functional equations in Schwartz space and their applications

Abstract

In the paper we consider functional equations of form $$ (B+r^{2}E)u(z)=0, $$ where $B$ is a constant complex $n\times n$ matrix, $E$ is the unit $n\times n$ matrix, $z$ is a complex variable, $r=|z|$, $u(z)$ is the sought generalized vector function. For this equation, we study the existence of non-trivial solutions and the manifold of all solutions in the functional space $D'=D'(\mathbb{C},\mathbb{C}^{n})$ of generalized vector function and in the space $S'=S'(\mathbb{C},\mathbb{C}^{n})$ of tempered distributions. We also study the existence of solutions growing at most polynomially at infinity. Such study is motivated by the problem on finding the solutions in $S'$ for elliptic systems of first order elliptic equations. Here an important role is played by the statement on the structure of distributions supported in a circumference. This statement provides an explicit representation of distributions supported in a circumference and this representation consists of a linear combinations of Cartesian product of periodic distributions and $\delta$-function and its derivatives. The process of finding all solutions to this equation in the space $D'$ consists of three stages. At the first stage, by reducing the matrix to the normal Jordan form, we split this equation into one-dimensional equations. At the second stage we prove that if the matrix $B$ has non negative and zero eigenvalues, that is, $\sigma(B)\cap(-\infty,0]=\varnothing$, where $\sigma(B)$ is the spectrum of the matrix $B$, then in the space $D'$, this equation has only the trivial solution. At the third stage, in the case $\sigma(B)\cap(-\infty,0]\neq\varnothing,$ we find all solutions to this equation in the space $D'$. Subject to the eigenvalues of the matrix $B$, the set of all solutions to this equation in the space $D'$ is either zero or depends on finitely many arbitrary $2\pi$-periodic distributions of one variable and finitely many arbitrary constants. The number of these functions and constants depend on the order of the solution; the order is prescribed. As an application, we find solutions in the space $S'$, in particular, polynomially growing solutions to elliptic systems of partial differential equations and to overdetermined systems. The results obtained in the work can be employed in studying the problems on solutions defined on the entire complex plane or a half-plane and in studying more general linear multi-dimensional elliptic systems and overdetermined systems of partial differential equations.