Boundary-value problems with singular conditions on boundary components of small dimensions (Q2754842)

The author considers boundary-value problems for the Laplacian on the domain \(\Omega \setminus F\), where \(\Omega \subset \mathbb R^3\) is a bounded domain with smooth boundary, \(F\subset \Omega\) is either one point, or a smooth curve. The boundary conditions are of the Dirichlet or Neumann type on \(\partial \Omega\) and singular ones on \(F\), that is, they contain weights vanishing on \(F\). Thus a solution may have singularities on \(F\) of prescribed types. Criteria for unique solvability are obtained. A similar approach to parabolic problems is also indicated.