Artenkunde des Gesprächs bei Alexander Kluge

Kac regular sets and Sobolev spaces in geometry, probability and quantum physics

“…It is shown in [13] that in the Euclidean case Ω's with ∂Ω having the segment property are Kac regular. In fact, many comparable Euclidean results can be found in [26]. The segment property is more general then being locally Lipschitz, but is also a concept that does not apply to manifolds.…”

“…(quasi everywhere) is understood with respect to the capacity associated with the regular strongly local Dirichlet form (11). A counterexample to this conjecture was given in [12] and [26] (cf. [27], p.127), who even show that there exists Ω ⊂ R m open with R m \ Ω compact and R m \ Ω =R m \ Ω such that (10) and thus (11) fails.…”