Solvability and smoothness of solution to variational Dirichlet problem in entire space associated with a non-coercive form

Abstract

In the work we study the solvability of the variational Dirichlet problem for one class of higher order degenerate elliptic operators in an entire $n$-dimensional Euclidean space. The coefficients of the operator have a power-law degeneracy at the infinity. The formulation of the problem is related with integro-differential sesquilinear form, which may not satisfy the coercivity condition. Earlier, the variational Dirichlet problem for degenerate elliptic operators associated with noncoercive forms was studied mostly for a bounded domain by means of a method based on a finite partition of unity of the domain. In contrast to this, we employ a special infinite partition of unity of the entire Euclidean space of finite multiplicity. The method used is based on techniques from the theory of spaces of differentiable functions of many real variables with a power weight. The boundary conditions in the problem are homogeneous in the sense that a solution to the problem is sought in a functional space in which the set of infinitely differentiable compactly supported functions is dense. The differential operator depends on the complex parameter $\lambda$, and the existence and uniqueness of a solution of the variational Dirichlet problem is proved in the case as $\lambda$ belongs to a certain angular sector with a vertex at zero that contains the negative part of the real axis. Under additional conditions on the smoothness of the coefficients and the right-hand side of the equation, the differential properties of the solution are studied.