Abstract
Let be a complete Riemannian manifold of dimension , let be a measure on with density with respect to the Riemannian volume, and let , where and . It is shown that in the case and the operator on the domain has a unique extension generating a -semigroup on , that is, the set is dense in . In particular, the operator is essentially self-adjoint on . A similar result is proved for elliptic operators with non-constant second order part that are formally symmetric with respect to some measure.
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