On the Dimension of the Solution Space of Elliptic Systems in Unbounded Domains

On the Mixed Dirichlet–Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

“…As a rule, to this end, one usually poses either the condition that the Dirichlet (energy) integral is finite or a condition on the character of vanishing of the modulus of the solution as |x| → ∞. Such conditions at infinity are natural and were studied by several authors (e.g., [1][2][3]).…”

On solutions of the Dirichlet problem for the polyharmonic equation in unbounded domains

“…In [3][4][5][6][7], generalizations of the Hardy inequality were established for bounded and for a wide class of unbounded domains; these inequalities were used to investigate boundary value problems for elliptic equations and systems. In particular, the problems of the existence, the uniqueness, the stability and the asymptotic expansions of solutions of boundary value problems were studied under the assumption that the energy (Dirichlet) integral is finite.…”

“…Developing an approach based on Hardy type inequalities [3], [5][6][7], in the present paper we succeed to obtain a uniqueness (non-uniqueness) criterion for solution of the Dirichlet problem for the polyharmonic equation in the exterior of a compact and in a half-space. To construct the solution, we use a variational method, that is, we minimize the corresponding functional in the class of admissible functions.…”

Smoothness of solutions of the Dirichlet problem for the biharmonic equation in nonsmooth 2D domains

“…Moreover, energy estimates expressing the Saint-Venant principle in elasticity theory for an arbitrary flat body were obtained, and uniqueness theorems for the Dirichlet problem in an unbounded domain were proved in a function class depending on the geometry of the domain. Finally, sharp estimates characterizing the behavior of |u(x)| and |∇u(x)| as |x| → ∞ were obtained.The uniqueness of solutions of boundary value problems for higher-order elliptic systems in unbounded domains with the behavior of solutions on infinity constrained by the condition of finiteness of the Dirichlet integral was studied in [6].The uniqueness of solutions was studied and the dimensions of the solution spaces of boundary value problems were found in [7]-[9] for the system of elasticity equations and the biharmonic (as well as polyharmonic) equation in various classes of unbounded domains with finite weighted energy (Dirichlet) integral.The behavior as |x| → ∞ of generalized solutions of the Dirichlet problem for the Navier-Stokes equations and the von Karman system in a neighborhood of the point at infinity was considered in [10], and the smoothness of generalized solutions of the Dirichlet problem for the Navier-Stokes equations in nonsmooth two-dimensional domains was studied in [11]. These papers also provide estimates characterizing the behavior of solutions of the Dirichlet problem under certain geometric conditions imposed on the boundary of the domain.…”

“…The uniqueness of solutions of boundary value problems for higher-order elliptic systems in unbounded domains with the behavior of solutions on infinity constrained by the condition of finiteness of the Dirichlet integral was studied in [6].…”