Sets Of Finite Perimeter Research Articles

A characterization of ℓ1 double bubbles with general interface interaction

Abstract We investigate the optimal arrangements of two planar sets of given volume which are minimizing the ℓ 1 {\ell_{1}} double-bubble interaction functional. The latter features a competition between the minimization of the ℓ 1 {\ell_{1}} perimeters of the two sets and the maximization of their ℓ 1 {\ell_{1}} interface. We investigate the problem in its full generality for sets of finite perimeter, by considering the whole range of possible interaction intensities and all relative volumes of the two sets. The main result is the complete classification of minimizers.

Bifurcation for a sharp interface generation problem

As opposed to the widely studied bifurcation phenomena for maps or PDE problems, we are concerned with bifurcation for stationary points of a nonlocal variational functional defined not on functions but on sets of finite perimeter, and involving a nonlocal term. This sharp interface model (1.2), arised as the \Gamma -limit of the FitzHugh–Nagumo energy functional in a (flat) square torus in {\mathbb{R}}^{2} of size T , possesses lamellar stationary points of various widths with well-understood stability ranges and exhibits many interesting phenomena of pattern formation as well as wave propagation. We prove that when the lamella loses its stability, bifurcation occurs, leading to a two-dimensional branch of nonplanar stationary points. Thinner nonplanar structures, achieved through a smaller T , or multiple layered lamellae in the same-sized torus, are more stable. To the best of our knowledge, bifurcation for nonlocal problems in a geometric measure theoretic setting is an entirely new result.

Quantitative C 1-stability of spheres in rank one symmetric spaces of non-compact type

Abstract We prove that in any rank one symmetric space of non-compact type M ∈ { ℝ ⁢ H n , ℂ ⁢ H m , ℍ ⁢ H m , 𝕆 ⁢ H 2 } {M\in\{\mathbb{R}H^{n},\mathbb{C}H^{m},\mathbb{H}H^{m},\mathbb{O}H^{2}\}} , geodesic spheres are uniformly quantitatively stable with respect to small C 1 {C^{1}} -volume preserving perturbations. We quantify the gain of perimeter in terms of the W 1 , 2 {W^{1,2}} -norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in M. As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter.

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>

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.

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.

Fractional differential operators, fractional Sobolev spaces and fractional variation on homogeneous Carnot groups

We establish a novel definition of the fractional gradient and divergence of order $$\alpha \in (0, 1)$$ through the use of Riesz potential on homogeneous Carnot groups. We introduce and investigate the distributional fractional Sobolev space and the space of fractional BV functions in this context. Additionally, we provide a definition of fractional Caccioppoli sets on homogeneous Carnot groups and demonstrate their blow-up property, using similar methods as outlined in [13].

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 ).

The Gauss-Green theorem for bounded vectorfields with divergence measure on sets of finite perimeter

The Gauss-Green theorem for bounded vectorfields with divergence measure on sets of finite perimeter

The asymptotics of the area-preserving mean curvature and the Mullins–Sekerka flow in two dimensions

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins–Sekerka flow.

A Fine Property of Sets at Points of Lebesgue Density

Consider an integer n≥2, m∈[n,+∞) and put α‾:=m−1n−1. Moreover: ∙ Let γx:(0,+∞)→Rn be the line through x=(x1,…,xn)∈Rn defined as γx(t):=(tx1,tα‾x2,…,tα‾xn); ∙ If T is any motion of Rn, then let γx(T) be the line through x=(x1,…,xn)∈Rn defined as γx(T):=T∘γT−1(x); ∙ Let H1 denote the one-dimensional Hausdorff measure in Rn. The main goal of this paper is to prove the following property: If x0 is an m-density point of a Lebesgue measurable set E and T is an arbitrary motion of Rn mapping the origin to x0, then we have lim supt→0+ H1(E∩γx(T)((0,t])) H1(γx(T)((0,t]))=1 for almost every x∈T({1}×Rn−1). An application of this result to locally finite perimeter sets is provided.