Let Ω be an open subset of R d and H Ω - Σ d i,j 1 ∂ iij ∂ j be a second-order partial differential operator on L 2(Ω) with domain C ∞ 0(Ω), where the coefficients c ij ε W 1,∞(Ω) are real symmetric and C = (c ij) is a strictly positive-definite matrix over Ω. In particular, H Ω is locally strongly elliptic. We analyze the submarkovian extensions of H Ω,i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that H Ωis Markov unique, i.e., it has a unique submarkovian extension, if and only if cap Ω(∂Ω) 0 where cap Ω (∂Ω) is the capacity of the boundary of Ω measured with respect to H Ω. The second main result shows that Markov uniqueness of H Ω is equivalent to the semigroup generated by the Friedrichs extension of H Ω being conservative.
|Number of pages||28|
|Journal||Annali della Scuola Normale - Classe di Scienze|
|Publication status||Published - 2011|