Markovian extensions of symmetric second order elliptic differential operators

Abstract

Key words Self-adjoint extensions, Markovian operators, elliptic differential operators MSC (2010) Primary, 47B25, Secondary, 47B38, 35J25Let ⊂ R n be bounded with a smooth boundary and let S be the symmetric operator in L 2 ( ) given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of S by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms in L 2 ( ) which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kreȋn's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.

Boundary representation of Dirichlet forms on discrete spaces

“…The present paper is a continuation of this theme building up on the results in [GHK + 15,KLSW17] and presenting further confirmation for this point of view. More specifically, our main result provides an analogue to the result of Posilicano [Pos14] mentioned above and provides a one-to-one correspondence of the Dirichlet forms on a graph and the Dirichlet forms on the Royden boundary.…”

“…In this context is it worth emphasizing that our approach is quite different from the one taken in [Pos14]. Indeed, right from the outset our situation is different as there is no geometrical boundary available.…”

“…For special such Markovian extensions (whose semigroups lie between the Dirichlet-and the Neumann semigroup) an explicit description in terms of boundary conditions was then given by Arendt and Warma [AW03] in 2003. An explicit description of all Markovian extensions in terms of boundary conditions or rather Dirichlet forms (in the wide sense) on the boundary could only recently be given by Posilicano [Pos14]. His approach relies on connecting self-adjoint extensions and boundary conditions via Krein's resolvent formula; a topic with revived interest in recent years [Pos08, Ryz07, BMNW08, PR09, Gru08, BGW09, Mal10, GM11].…”

Nonlinear maximal monotone extensions of symmetric operators

“…Here a ∧ b := min{a, b} and a ∨ b := max{a, b}. Then, proceeding as in the linear case considered in [28], by γ 0 (f ∧ g) = γ 0 f ∧ γ 0 g and γ 0 (f ∨ g) = γ 0 f ∨ γ 0 g, one can check that…”

Asymptotics of semigroups generated by operator matrices

“…Actually, there is a long series of different but complementary results on Wentzell-type boundary conditions in the theory of Dirichlet forms, see e.g. [43] for an updated series of references.…”