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Register a new userCharacterizations of sets of finite perimeter using heat kernels in metric spaces
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The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with $N^{1,1}$-spaces) and the theory of heat semigroups (a concept related to $N^{1,2}$-spaces) in the setting of metric measure spaces whose measure is doubling and supports a $1$-Poincare inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of ${rm BV}$ functions in terms of a near-diagonal energy in this general setting.
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We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar{e} inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.
We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $Gamma$ that, similarly to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset $C$ whose translations by elements in $Gamma$ dont intersect $Gamma$. However, a priori the cone $C$ may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second type.
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.
We analyze a class of weakly differentiable vector fields (FF colon rn to rn) with the property that (FFin L^{infty}) and (div FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field $FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we establish a fundamental approximation theorem which states that, given a Radon measure $mu$ that is absolutely continuous with respect to $mathcal{H}^{N-1}$ on $rn$, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure $|mu|$. With this approximation theorem, we derive the normal trace of $FF$ on the boundary of any set of finite perimeter, (E), as the limit of the normal traces of $FF$ on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for $FF$ holds on (E). With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an $(N-1)$-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.
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