A local increase in the smoothness of generalized solutions to elliptic boundary value problem in nonsmooth domains (Q1374924)

Let \(G\subset \mathbb{R}^n\) be a bounded domain, and \(M\subset\partial G\) the set of singular points, \(\partial G\setminus M\in C^\infty\). We consider the following elliptic boundary problem in \(G\): \[ L(x,D)u=f \quad\text{in }G; \qquad \text{ord }L=2m; \tag{1} \] \[ B_j(x,D) u|_{\partial G\setminus M}= \varphi_j, \quad j=1,\dots,m; \qquad \text{ord }B_j= m_j. \tag{2} \] Assume that \(x_0\) belongs to a smooth open piece of a nonsmooth boundary. The question naturally arises whether the assertion on local raising of smoothness is valid in some neighbourhood \(U(x_0)\) of the point \(x_0\) in \(\overline{G}\); is this assertion true if \(x_0\) is an internal point of the domain? V. A. Kondrat'ev set up this problem in 1992. We deal here with the solution to this problem.