The Identity of the Weak and Strong Extensions of a Linear Elliptic Differential Operator: Ii.

be a linear differential operator in Q with indefinitely differentiable complex-valued coefficients aj,. ... X n) e Q. D operates on the space of distributions in U. Let Di denote the restriction of D to the space of C' functions with compact supports. The weak extension Dw of DI in L2(Q) is defined as follows: the domain M of Dw consists of all those elements f in L2 for which Df, formed in the sense of distributions, is in L2, and for such f, Dwf is defined to be Df. The strong extension Ds of Di in L2 is the closure in L2 of the operator D2, where D2 is defined as follows: its domain consists of all f in L2 which are C' and for which Df, formed in the usual sense, belongs to L2, and for such f, DJ = Df. It is easily seen that Dw is an extension of Ds. It is not known whether, in general, the weak and strong extensions coincide. In this note we prove their identity for a class of differential operators. 1 2. THEOREM 1. Let Q be an arbitrary open subset of R', and let D be formally self-adjoint and elliptic.2 The weak and strong extensions of Di in L2(Q) coincide. Proof: By assumption, Di is a symmetric operator in L2; also, Dw is the adjoint of Di in L2. We have, by a theorem of von Neumann,3