Generalized solutions of elliptic boundary value problems with strong power singularities (Q2735863)

The following problem is considered NEWLINE\[NEWLINEA(x,D)u= F_0,\;x\in \Omega, \quad B_j(x,D)u|_{\partial\Omega}= F_j\;(j= \overline{1,m})NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with boundary \(S\) of class \(C^\infty\), \(A(x,D)\) is an elliptic operator of order \(2m\), \(\{B_j (x,D)\}_{j=1}^m\), is some normal system of boundary differential expressions satisfying Lopatinsky's condition. The coefficients of the operators are infinitely differentiable. The author investigates the character of singularities of the solution on power of data singularities. With this purpose \(\forall\,k\in \mathbb{R}^1\) the spaces of functions \(Z_k (\overline{\Omega},x_0)\), \(x_0\in \overline{\Omega}\), NEWLINE\[NEWLINE Z_k(\overline{\Omega},x_0)= \{\varphi\in C^\infty (\overline{\Omega} \setminus x_0), p^{| \alpha|} (x,x_0) D^\alpha \varphi(x)= \rho^k(x,x_0) \varphi_\alpha(x),\;\varphi_\alpha\in C^\infty (\overline{\Omega}), \forall, \alpha\}NEWLINE\]NEWLINE are introduced, where \(\rho(x, x_0)=| x-x_0| \sigma(x, x_0)\), \(\sigma\in C_0^\infty(\Omega)\) is the cutting function. The space of linear continued functionals on \(Z_k (\overline{\Omega}, x_0)\) is denoted by \(Z_k' (\overline{\Omega}, x_0)\). It turns out \(Z_{-k} (\overline{\Omega}, x_0) \subset Z_k' (\overline{\Omega}, x_0)\).NEWLINENEWLINE The main result: Let \(F_0\in Z_p' (\overline{\Omega}, x_0)\), \(p> 2m-n\), \(F_1=\cdots= F_m= 0\), kerned of the adjoint problem \(N^*= \{0\}\) and \(u(x)\) is a solution of problem (1) which is orthogonal to its kernel. Then \(u(x)\in Z_{p-2m}' (\overline{\Omega}, x_0)\).NEWLINENEWLINE In the proof the existence of a normal fundamental solution is assumed. There are some misprints.