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

Abstract

Let Ω ⊂ R 2 be a bounded domain whose boundary ∂Ω contains the origin 0, let Ω = Ω ∪ ∂Ω be the closure of Ω, let x = (x 1 , x 2 ), and let |x| = x 2 1 + x 2 2 . Consider the Dirichlet problem for the biharmonic equationin Ω with the boundary conditionsThe asymptotics, smoothness, and uniqueness issues and methods for studying them in such problems were developed by Kondrat'ev, Oleinik, and their students [1]- [9]. In particular, estimates for the solution of the Dirichlet problem for the biharmonic equation in a neighborhood of irregular boundary points under weak assumptions on the structure of the boundary in the vicinity of these points, as well as estimates characterizing the solution behavior in a neighborhood of irregular boundary points, were obtained in [1]- [5]. The behavior of a generalized solution of the Dirichlet problem for the biharmonic equation in a neighborhood of a boundary point and in a neighborhood of infinity in the case of two independent variables was studied there. 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.In the present paper, we use the method of weight functions to study the smoothness of solutions of the Dirichlet problem for the biharmonic equation in a two-dimensional domain with nonsmooth boundary. We introduce the following notation: *