Finite Perimeter Research Articles

Generic Level Sets in Mean Curvature Flow are BV Solutions

We show that a generic level set of the viscosity solution to mean curvature flow is a distributional solution in the framework of sets of finite perimeter by Luckhaus and Sturzenhecker, which in addition saturates the optimal energy dissipation rate. This extends the fundamental work of Evans and Spruck (J Geom Anal 5(1):79–116, 1995), which draws a similar connection between the viscosity solution and Brakke flows.

Minimizing movements for forced anisotropic curvature flow of droplets

We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in Bellettini and Kholmatov [J. Math. Pures Appl. 117 (2018), 1–58], we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young’s law, and also the existence of a 1/2 -Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness, and the consistency with the smooth flow.

Set-decomposition of normal rectifiable $G$-chains via an abstract decomposition principle

We introduce the notion of set-decomposition of a normal G -flat chain A in \mathbb{R}^{n} as a sequence A_{j}=A\,\text{\Large{$\llcorner$}}S_{j} associated to a Borel partition S_{j} of \mathbb{R}^{n} such that \mathbb{N}(A)=\sum \mathbb{N}(A_{j}) . We show that any normal rectifiable G -flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their “measure theoretic” connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer.As opposed to previous results, we do not assume that G is boundedly compact. Therefore, we cannot rely on the compactness of sequences of chains with uniformly bounded \N -norms. We deduce instead the result from a new abstract decomposition principle.As in earlier proofs, a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h -mass to replace integrality.

On the Relaxation of Gauss’s Capillarity Theory Under Spanning Conditions

We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely “wet" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class.

Sobolev, BV and perimeter extensions in metric measure spaces

We study extensions of sets and functions in general metric measure spaces. We show that an open set has the strong BV-extension property if and only if it has the strong extension property for sets of finite perimeter. We also prove several implications between the strong BV-extension property and extendability of two different non-equivalent versions of Sobolev \(W^{1,1}\)-spaces and show via examples that the remaining implications fail.

A Hessian-Dependent Functional With Free Boundaries and Applications to Mean-Field Games

We study a Hessian-dependent functional driven by a fully nonlinear operator. The associated Euler-Lagrange equation is a fully nonlinear mean-field game with free boundaries. Our findings include the existence of solutions to the mean-field game, together with Hölder continuity of the value function and improved integrability of the density. In addition, we prove the reduced free boundary is a set of finite perimeter. To conclude our analysis, we prove a Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence result for the functional.

On the properties of the set where a generalized function of bounded variation takes an infinite value

We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim, we prove a new result on strict approximation of the sets of finite perimeter from the outside with open sets.

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

<p style='text-indent:20px;'>We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.</p>

A necessary condition for Sobolev extension domains in higher dimensions

We give a necessary condition for a domain to have a bounded extension operator from L1,p(Ω) to L1,p(Rn) for the range 1<p<2. The condition is given in terms of a power of the distance to the boundary of Ω integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for W1,1-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain in R3 that is homeomorphic to a ball and has 3-dimensional boundary.

Embeddedness of liquid-vapour interfaces in stable equilibrium

We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g., water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g., air), it may come into contact with solid supports (e.g., a container), and it is subject to the action of an analytic potential field (e.g., gravity). The region occupied by the liquid is described as a set of locally finite perimeter (Caccioppoli set) in \R^3 ; no a priori regularity assumption is made on its boundary. The (twofold) scope in this note is to propose a weakest possible set of mathematical assumptions that sensibly describe a condition of stable equilibrium for the liquid-vapour interface (the capillary surface), and to infer from those that this interface is a smoothly embedded analytic surface. (The liquid-solid-vapour junction, or free boundary, can be present but is not analysed here.) The result relies fundamentally on the recent varifold regularity theory developed by the author and Wickramasekera, and on the identification of a suitable formulation of the stability condition.

On the Gaussian Projection Inequalities

Abstract We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection-type inequality. This inequality concludes that Ehrhard symmetrization contracts the Minkowski symmetrization of the Gaussian projection body for set of finite Gaussian perimeter $E$. On the other hand, we investigate the functional “lifting” of Ehrhard symmetrization and establish the affine Gaussian Pólya–Szegö-type inequalities in terms of the functional Ehrhard symmetrization.

Monotone sets and local minimizers for the perimeter in Carnot groups

Monotone sets have been introduced about ten years ago by Cheeger and Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg group into L1 to the classification of its monotone subsets. Later on, monotone sets played an important role in several works related to geometric measure theory issues in the Heisenberg setting. In this paper, we work in an arbitrary Carnot group and show that its monotone subsets are sets with locally finite perimeter that are local minimizers for the perimeter. Under an additional condition on the ambient Carnot group, we prove that their measure-theoretic interior and support are precisely monotone. We also prove topological and measure-theoretic properties of local minimizers for the perimeter whose interest is independent from the study of monotone sets. As a combination of our results, we get in particular a sufficient condition under which any monotone set admits measure-theoretic representatives that are precisely monotone.

Ground states for a nonlocal isoperimetric problem with two interacting phases

ABSTRACT A nonlocal isoperimetric problem of two interacting phases with power-law potentials is investigated. Using the concentration-compactness lemma for sets of finite perimeter, which was introduced by Frank and Lieb, we prove the existence of ground states for the nonlocal isoperimetric problem with two interacting phases. Our result improve some of Frank and Lieb RL Frank, EH Lieb. A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J Math Anal. 2015;47(6):4436–4450. doi:10.1137/15M1010658. Moreover, as the potential function vanishes, we also obtain a necessary and sufficient condition for the existence of this nonlocal isoperimetric problem.

Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball

Abstract In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most n - 3 {n-3} . With this inequality, we classify all volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth.

Hypersurfaces with mean curvature prescribed by an ambient function: Compactness results

We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is gν, for a given continuous positive ambient function g, and where ν denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearance of “hidden boundaries”, that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that g does not prescribe the mean curvature in the limit. As a special instance of a more general result, we prove that locally uniform Lq-bounds on the (weakly defined) second fundamental form, for q>1, in addition to the customary locally uniform bounds on the perimeters, lead to a compact class of boundaries with mean curvature prescribed by g.The proof relies on treating the boundaries as oriented integral varifolds, in order to exploit their orientability feature (that is lost when treating them as (unoriented) varifolds). Specifically, it relies on the formulation and analysis of a weak notion of curvature coefficients for oriented integral varifolds (inspired by Hutchinson's work [7]).This framework gives (with no additional effort) a compactness result for oriented integral varifolds with curvature locally bounded in Lq with q>1 and with mean curvature prescribed by any g∈C0 (in fact, the function can vary with the varifold for which it prescribes the mean curvature, as long as there is locally uniform convergence of the prescribing functions). Our notions and statements are given in a Riemannian manifold, with the oriented varifolds that need not arise as boundaries (for instance, they could come from two-sided immersions).

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non‐smooth coefficients

AbstractWe prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C1‐boundary, the other stated for sets with finite perimeter.

Optimal control of moving sets

Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets t↦Ω(t)⊂IR2. Given an initial set Ω0, the goal is to minimize the area of the contaminated set Ω(t) over time, plus a cost related to the control effort. Here the control function is the inward normal speed along the boundary ∂Ω(t). We prove the existence of optimal solutions, within a class of sets with finite perimeter. Necessary conditions for optimality are then derived, in the form of a Pontryagin maximum principle. Additional optimality conditions show that the sets Ω(t) cannot have certain types of outward or inward corners. Finally, some explicit solutions are presented.

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $$\mathbb {T}^N$$ starting near a strictly stable critical set E of the perimeter converges in the long time to a translate of E exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

On smooth interior approximation of sets of finite perimeter

In this paper, we prove that for any bounded set of finite perimeterΩ⊂Rn\Omega \subset \mathbb {R}^n, we can choose smooth setsEk⋐ΩE_k \Subset \Omegasuch thatEk→ΩE_k \rightarrow \OmegainL1L^1andlim supi→∞P(Ei)≤P(Ω)+C1(n)Hn−1(∂Ω∩Ω1).\begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega )+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*}In the aboveΩ1\Omega ^1is the measure-theoretic interior ofΩ\Omega,P(⋅)P(\cdot )denotes the perimeter functional on sets, andC1(n)C_1(n)is a dimensional constant.Conversely, we prove that for any setsEk⋐ΩE_k \Subset \OmegasatisfyingEk→ΩE_k \rightarrow \OmegainL1L^1, there exists a dimensional constantC2(n)C_2(n)such that the following inequality holds:lim infk→∞P(Ek)≥P(Ω)+C2(n)Hn−1(∂Ω∩Ω1).\begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega )+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*}In particular, these results imply that for a bounded setΩ\Omegaof finite perimeter,Hn−1(∂Ω∩Ω1)=0\begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*}holds if and only if there exists a sequence of smooth setsEkE_ksuch thatEk⋐ΩE_k \Subset \Omega,Ek→ΩE_k \rightarrow \OmegainL1L^1andP(Ek)→P(Ω)P(E_k) \rightarrow P(\Omega ).

Rigidity for measurable sets

Let Ω⊂Rd, d≥2, be a set with finite Lebesgue measure such that, for a fixed radius r>0, the Lebesgue measure of Ω∩Br(x) is equal to a positive constant when x varies in the essential boundary of Ω. We prove that Ω is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than r which is either open and connected, or of finite perimeter and indecomposable. The proof requires reinventing each step of the moving planes method by Alexandrov in the framework of measurable sets.