Sentence examples for of finite perimeter from inspiring English sources

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter.

The main result establishes convergence to a weak formulation of (multi-phase) mean curvature flow in the BV-framework of sets of finite perimeter.

We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e.

We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter.

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure.

Let E be a set of finite perimeter.

Though most of the results that we are going to present could be given first for smooth sets and then extended to sets of finite perimeters via the approximation Theorem 2.4 below, we have preferred to state them in full generality.

We say that a set of locally finite perimeter E is an area ((K, r_0)) -quasi-minimizer if for every F, such that ( F Delta E subset subset B_{r}(x)), with (r le r_0), the following inequality holds begin{aligned} P(E; B_{r}(x) ) le K, P F; B_{r}(x)).

If is -rectifiable, then it has locally finite perimeter in the sense of De Giorgi, and therefore a unit normal vector exists -almost everywhere on [7, Sections 3.2.14, 3.2.15].

On the other hand, a set in ({mathcal {C}}_r) has always finite perimeter [73, Prop. 2.4], but the ({mathcal H}^{n-1} -measure of its topological boundary may be strictly larger tH}^{n-1} -measure, [73, Example 2.6].

Subsequently, the same formula was proved in [32] and [7] under additional smoothness assumptions on E. On the other hand, as a consequence of [98, Th. 3], we have that, for any set E of finite measure and finite s-perimeter for all (sin (0,1)), begin{aligned} lim _{sdownarrow 0},sP_s(E =nomega _n,|E|.