We study a system of fully nonlinear elliptic equations, depending on a small parameter $$\varepsilon $$ , that models long-range segregation of populations. The diffusion is governed by the negative Pucci operator. In the linear case, this system was previously investigated by Caffarelli, Patrizi, and Quitalo in J. Eur. Math. Soc. 19, 3575–3628 (2017) as a model in population dynamics. We establish the existence of solutions and prove convergence as $$\varepsilon \rightarrow 0^+$$ to a free boundary problem in which populations remain segregated at a positive distance. In addition, we show that the supports of the limiting functions are sets of finite perimeter and satisfy a semi-convexity property.

The classical Serrin’s overdetermined theorem states that a C^{2} bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin’s theorem to Lipschitz domains remained unresolved. Our paper answers this open question affirmatively. Actually, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.

In this paper, we give a decomposition of the gradient measure [Formula: see text] of an arbitrary function of bounded variation [Formula: see text] into a linear combination of atoms [Formula: see text], where [Formula: see text] is a set of finite perimeter. The atoms further satisfy the support, cancellation, normalization, and size conditions: For each [Formula: see text], there exists a cube [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], and, denoting by [Formula: see text] the heat kernel in [Formula: see text], [Formula: see text] Our proof relies on a sampling of the coarea formula and a new boxing identity. We present several consequences of this result, including Sobolev inequalities, dimension estimates, and trace inequalities.

Abstract We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds). As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant. The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C k C^{k} -norms. Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres.

Abstract We consider a CMC hypersurface with an isolated singular point at which the tangent cone is regular, and such that, in a neighbourhood of said point, the hypersurface is the boundary of a Caccioppoli set that minimises the standard prescribed-mean-curvature functional. We prove that in a ball centred at the singularity there exists a sequence of smooth CMC hypersurfaces, with the same prescribed mean curvature, that converge to the given one. Moreover, these hypersurfaces arise as boundaries of minimisers. In ambient dimension 8 the condition on the cone is redundant. (When the mean curvature vanishes identically, the result is the well-known Hardt–Simon approximation theorem.

In this paper, we aim to develop the foundations of a theory of BV functions in the configuration space over the Euclidean space Rn equipped with the Poisson measure π. We first construct the m-codimensional Poisson measure—formally written as “(∞-m)-dimensional Poisson measure”—on the configuration space. We then show that our construction is consistent with potential theory induced by the infinitely many independent Brownian motions by establishing relations between the m-codimensional Poisson measure and Bessel capacities. Secondly, we introduce three different definitions of BV functions based on the variational, relaxation, and semigroup approaches, and prove the equivalence of them. In the process, we prove the p-Bakry–Émery inequality on the configuration space for any 1<p<∞. Thirdly, we construct perimeter measures and introduce an appropriate notion of measure-theoretic boundary, called the reduced boundary. We then prove that the perimeter measure can be expressed by the 1-codimensional Poisson measure restricted to the reduced boundary, which is a generalisation of De Giorgi’s identity to the configuration space. Finally, we construct the total variation measures for functions of bounded variation, and prove the Gauß–Green formula.

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.

The purpose of this paper is to investigate the Almgren problem in R3 under generic conditions on the potential and tension functions which define the free energy. This problem appears in classical thermodynamics when one seeks to understand whether minimizing the free energy with convex potential in the class of sets of finite perimeter under a mass constraint generates a convex minimizer representing a crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize convexity and a stability theorem when the mass is small, as well as a first-variation partial differential equation along with a new maximum principle approach.

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.

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.

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.

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.

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.

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.

We investigate a phase-field version of the Faber–Krahn theorem based on a phase-field optimization problem introduced by Garcke et al. in their 2023 paper formulated for the principal eigenvalue of the Dirichlet–Laplacian. The shape that is to be optimized is represented by a phase-field function mapping into the interval [0,1] . We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to 0 and 1 except for a thin transition layer whose thickness is of order \varepsilon&gt;0 . Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a \Gamma -convergence result which allows us to recover a variant of the Faber–Krahn theorem for sets of finite perimeter in the sharp interface limit.

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.

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.

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.

&lt;p style='text-indent:20px;'&gt;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.&lt;/p&gt;

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.