Properties of a solution of the Dirichlet problem for second-order elliptic equations in a class of domains with nonsmooth boundary. I, II (Q1089503)

This paper is devoted to the study of the differential properties of a solution of a Dirichlet problem for the second-order elliptic equation \[ {\mathcal L}u\equiv \sum^{n}_{i,j=1}(\partial /\partial x_ i)(a_{ij}u_{x_ j}+a_ iu)+\sum^{n}_{i=1}b_ iu_{x_ i}+au=f,\quad u|_{\partial \Omega}=0, \] in a bounded domain \(\Omega \subset {\mathbb{R}}^ n\), \(n\geq 3\). It is assumed that the boundary \(\partial \Omega\) of the domain \(\Omega\) consists of two closed smooth (i.e., infinitely differentiable) surfaces \(\Gamma_ 1\) and \(\Gamma_ 2\) such that the set \(\Pi_ 0=\Gamma_ 1\cap \Gamma_ 2\) is nonempty and is a closed smooth manifold of dimension n-2, and there exists a small number \(d_ 0>0\) such that for any point \(\xi \in \Pi_ 0\) the part \(\Omega_ d(\xi)\) of the domain \(\Omega\) located in the ball \(B_ d(\xi)=\{x\in {\mathbb{R}}^ n:| x-\xi | <d\}\), \(d\in (0,d_ 0)\) is mapped with the help of a nondegenerate diffeomorphism \(T_{\xi}\) of class \(C^{\infty}\) into some subdomain \(U_ d(\xi)\) of the domain \[ {\mathcal D}=\{y\in {\mathbb{R}}^ n: | y_ 1| <g(y_ 2),(y^ 2,\hat y)\in R^ 1_+\times {\mathbb{R}}^{n-2},\quad \hat y=(y_ 3,....,y_ n)\}, \] which we shall call the dihedral angle of opening zero.