The boundary conditions description of type i domains

Abstract

Type I domains are the domains of the self-adjoint operators determined by the weak formulation of formally self-adjoint differential expressions l. This class of operators is defined by the requirement that the sesquilinear form q(u, v) obtained from l by integration by parts agrees with the inner product < lu, v >. A complete characterisation of the boundary conditions assumed by functions in these domains for second-order differential expressions is given in this paper. In the singular case, the boundary conditions are stated in terms of certain 'boundary condition' functions and in the regular case they are given in terms of classical function values.