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On essential self-adjointness and stochastic completeness for second order elliptic PDO in $\Omega \subset {\mathbb R}^n$

Gheorghe Nenciu
(Institute of Mathematics of the Romanian Academy)
Tirsdag, 7 oktober, 2014, at 17:15-18:00, in Koll. G4 (1532-222)
Let $\Omega$ be a connected domain in ${\mathbb R}^n$ and 

$H_V = -\frac{1}{\mu(x)} \sum_{j,k=1}^n \partial_j \mu(x) D_{j,k}(x) \partial_k + V(x),$

${\mathcal D}(H_V) = C_{0}^{\infty}(\Omega)$. The matrix ${\mathbf D}(x)$ is real, symmetric and strictly positive, $\mu(x)>0$ and $V(x) = \overline{V(x)}$.

We discuss conditions on the behaviour of  ${\mathbf D}(x), \mu(x), V(x)$ near the boundary $\partial \Omega$ of $\Omega$, which insures that:
  1. $H_V$ is essentially self-adjoint.
  2. $H_{V=0}$ is stochastically complete, i.e. the semigroup generated by its Friedrichs extension is conservative.
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