A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

We present a new method for computing volume integrals based on data sampled on a regular cartesian grid. We treat the case where the domain is defined implicitly by a trivariate inequality f(x,y,z) < 0, and the input data includes sampled values of the defining function f and the integrand. The method employs Federer’s coarea formula to convert the volume integral to an integral over level set values where the integrand is an integral over the level sets. Application of a standard quadrature method produces an approximation of the integral over the continuous range of f in the form of a sum of integrals on level sets corresponding to a discrete set of values of f. The integrals on the discrete collection of level sets are evaluated using the grid-based approach presented by Yurtoglu et al. [1]. We describe how the implementation relates to the implementation described in [1], and we present results for sample problems with known exact results to support a discussion of accuracy and convergence along with a comparison with a traditional Monte Carlo method.

Functions of bounded higher variation in the fractional Sobolev spaces

In this paper, we establish fine properties of functions of bounded higher variation in the framework of fractional Sobolev spaces. In particular, inspired by the recent work of Brezis–Nguyen on the distributional Jacobian, we extend the definition of functions of bounded higher variation, which defined by Jerrard–Soner in [Formula: see text], to the fractional Sobolev space [Formula: see text], and apply Cartesian currents theory to establishing general versions of coarea formula, chain rule and decomposition property.

On Perimeters and Volumes of Fattened Sets

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a). We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(r). These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

Fractional Perimeters from a Fractal Perspective

The purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake{S\subseteq\mathbb{R}^{2}}this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as{s\to 1^{-}}of the fractional perimeter of a set having locally finite (classical) perimeter.

The L1L^{1} gradient flow of a generalized scale invariant Willmore energy for radially non-increasing functions

We use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L^{1}(\mathbb{R}^{d}). This functional is extended to every {L^{1}} function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation.