Singularities of a solution on a boundary of lower dimension
Suppose that \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with boundary \(\Gamma_{n-1}\cup \Gamma_{n-2}\), where \(\Gamma_{n-1}\) and \(\Gamma_{n-2}\) are \((n-1)\)-dimensional and \((n-2)\)-dimensional, respectively, smooth manifolds, \(\Gamma_{n-1}\) and \(\Gamma_{n-2}\) do not intersect, \(n>2\), and \(P(x,D)\) is a linear differential operator in \(\Omega\) with smooth coefficients on \(\Omega\cup\Gamma_{n-2}\), \(\deg P(x,D)= m\), \(m\geq 2\). We consider the following problem: \[ P(x, D)u(x)=0,\quad x\in\Omega,\quad u(x)= 0,\quad x\in\Gamma_{n-2}\backslash A,\tag{1} \] where \(A\) is a closed set strictly inside \(\Gamma_{n-2}\). We consider the singularity of the solution on the set \(A\), i.e., the case where \(u(x)=0\) on \(\Gamma_{n-2}\), if \(u(x)\) is the solution of problem (1).
scientific article; zbMATH DE number 1059807