J. Behrndt and M. Langer,
Boundary value problems for elliptic partial differential operators on bounded domains,
J. Funct. Anal. 243 (2007), 536–565

Abstract:
For a symmetric operator or relation A with infinite deficiency indices in a Hilbert space we develop an abstract framework for the description of symmetric and self-adjoint extensions AΘ of A as restrictions of an operator or relation T which is a core of the adjoint A. This concept is applied to second order elliptic partial differential operators on smooth bounded domains, and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated.

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