Sobolev functions on varifolds

Abstract

This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts.Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily Hölder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well.Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions. Mathematics Subject Classification 46E35 (primary), 49Q15, 53C22 (secondary). † Inner products are denoted by '•'; see [19, 1.7.1]. Moreover, whenever P is an m-dimensional plane in R n , the orthogonal projection of R n onto P is denoted by P ; see Almgren [3, T.1(9)]. † If μ measures a metric space X, a ∈ X, and m is a positive integer, then Θ m (μ, a) = lim r→0+ μ B(a, r) α(m)r m , where α(m) = L m B(0, 1) and B(a, r) is the closed ball with centre a and radius r; see [19, 2.7.16, 2.8.1, 2.10.19].‡ The spaces Lp(μ, Y ) and L loc p (μ, Y ) contain functions rather than equivalence classes of functions. Hom(R n , Y ))} equals the usual space of weakly differentiable functions, see [32, 8.18]. However, considering the varifold associated to three lines in R 2 meeting at a common point at equal angles shows that the indicated subclass need not to be closed with respect to addition; see [32, 8.The more elaborate properties of T(V, Y ) build on the isoperimetric inequality which works most effectively under the density hypothesis. This hypothesis allows for the formulation of various Sobolev-Poincaré-type inequalities with and without boundary condition; see [32, 10.1, 10.7, 10.9]. Furthermore, pointwise differentiability results both of approximate and integral nature then hold for generalised V weakly differentiable functions; see [32, 11.2, 11.4].Turning to some relevant results concerning the local connectedness structure of varifolds, the mean curvature hypothesis becomes more relevant. Under the density hypothesis and the mean † If U is an open subset of some finite-dimensional normed vectorspace and Z is a Banach space, then E (U, Z) denotes the vectorspace of functions θ : U → Z of class ∞.

Discretization and Approximation of Surfaces Using Varifolds

“…Here we recall the de nition of generalized curvature proposed by Hutchinson [16] (see also the recent reformulation due to Menne [20]).…”

“…Quite interestingly, it turns out that the boundary measure of a curvature varifold with boundary is (d − )-recti able and has an integral multiplicity (this follows from a very nice argument showing rst the local orientability of the varifold, and then applying Federer-Fleming's Integrality Theorem for currents). Moreover, the notion of curvature varifold has been shown to be equivalent to the so-called V-weak di erentiability of the approximate tangent map (see the recent work by Menne [20]).…”

Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds

“…In the present paper only products of varifolds with planes are employed. More general products will be required in the study of the geodesic distance on the support of the weight measure of certain varifolds, see [Men15b,§6].…”

A novel type of Sobolev-Poincaré inequality for submanifolds of Euclidean space

“…If we have p ≥ m in Hypothesis 2, then spt V is in many ways well-behaved. For instance, there holds Θ m * ( V , x) ≥ 1 for x ∈ spt V by [Men09, 2.7] -in particular, spt V has locally finite H m measure -, spt V is locally connected (see [Men16a,6.14 (3)]), decompositions of V are locally finite (see [Men16a,6.11]) and non-uniquely refine the decomposition of spt V into connected components (see [Men16a,6.13,6.14 (1)]), connected components of spt V are locally connected by paths of finite length (see [Men16a, 14.2]), and the resulting geodesic distance thereon is a continuous Sobolev functions with bounded generalised weak derivative (see [Men16b,6.8 (1)]).…”