A Saint-Venant type principle for Dirichlet forms on discontinuous media

Sch’nol’s theorem for strongly local forms

“…We should, moreover, mention our version of the Caccioppoli inequality, Theorem 3.1 below. For the unperturbed operator H 0 such an inequality can be found in [10]. Our version here, including measure perturbations, appears to be new and might be of interest in its own right.…”

“…Throughout we will work with a locally compact, separable metric space X endowed with a positive Radon measure m with suppm = X. Our exposition here goes pretty much along the same lines as those in [10,35]. We refer to [16] as the classical standard reference as well as [11,17,27,14] for literature on Dirichlet forms.…”

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

“…The first property is the doubling inequality for the measure of balls, balls defined using a control metric naturally associated to the operator, and the second property is the validity of a Poincaré inequality involving a notion of gradient naturally associated to the operator. This point of view, independently developed in the work of Biroli and Mosco [4] and Sturm [44], has been further studied by several authors, and has led to Harnack inequalities for more general classes of nonlinear parabolic PDE, see for instance [32], [33], [38], [39] and [6]. The ideas in [42] and [26], are based on Moser's approach [40].…”

Sobolev Inequalities in Familiar and Unfamiliar Settings

“…For instance, versions of the chain rule and Leibnitz rule apply. In what follows, we work under additional assumptions that imply that the set of those u in D(E) such that dΓ/dµ exists is rich enough (see [10,97] for further details). We now introduce a key ingredient to our discussion: the intrinsic distance.…”

“…Denote by B(x, r) the open balls in (M, ρ). Each B(x, r) is precompact with compact closure given by the associated closed [10,11,12,94,95,96,97] for details).…”