2012/11/30 by Andrea Posilicano
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Algebra over a field #Discrete mathematics #Mathematics #Nonlinear Partial Differential Equations #Operator (biology) #Order (exchange) #Pure mathematics #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #math.PR
paper · pdf · doi:10.1002/mana.201300277
Slightly revised version. Accepted for publication in Mathematische Nachrichten
arxiv created 2014/03/06 · openalex publication_date 2014/04/25 · arxiv updated 2014/06/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let be bounded with a smooth boundary Γ and let S be the symmetric operator in 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 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 Kreĭn'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.