We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $σ\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\p \{u>0\}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.
A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $Ω\subset\mathbb{R}^n$ open, bounded and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds \begin{equation*} λ_1(Ω)-λ_1(B)\geq c\left(P(Ω)-P(B) \right)^{2}, \end{equation*} where by $λ_1(\cdot)$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot)$ its perimeter. The hearth of the present pap…
We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $γ$ lower semicontinuous.
We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ Γ$-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $Γ$-converges to the classical perimeter, while for $a\in(0,1)$ the $Γ$-limit is a new nonlocal operator, tha…
This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior of sets having (locally) finite fractional perimeter and we establish existence and compactness results for (locally) $s$-minimal sets. We study the $s$-minimal sets in highly nonlocal regimes, that correspond to small values of the fractional parameter $s$. We introduce a functional framework for studying those $s$-minimal sets that can be globally written…
We establish the $Γ$-convergence of some energy functionals describing nonlocal attractive interactions in bounded domains. The interaction potential solves an elliptic equation (local or nonlocal) in the bounded domain and the primary interest of our results is to identify the effects that the boundary conditions imposed on the potential have on the limiting functional. We consider general Robin boundary conditions, which include Dirichlet and Neumann conditions as particular cases. Depending on the order of the elliptic operator the limiting functional involves the usual perimeter or some fr…
In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either…
We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function. These two nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing wh…
Inside a fixed bounded domain $Ω$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $λ_1(Ω\setminus K)$. We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and, in some cases non symmetry results, identifying the solution. We choose to work with the outer Minkowski content as the "good" notion of perimete…
We prove the existence of global minimizers to the double minimization problem \[ \inf\Big\{ P(E) + λW_p(\mathcal{L}^n \lfloor \, E,\mathcal{L}^n \lfloor\, F) \colon |E \cap F| = 0, \, |E| = |F| = 1\Big\}, \] where $P(E)$ denotes the perimeter of the set $E$, $W_p$ is the $p$-Wasserstein distance between Borel probability measures, and $λ> 0$ is arbitrary. The result holds in all space dimensions, for all $p \in [1,\infty),$ and for all positive $λ$. This answers a question of Buttazzo, Carlier, and Laborde.
In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian $λ$ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane…
Results are obtained for two minimization problems: $$I_k(c)=\inf \{λ_k(Ω): Ω \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(Ω)= c \},$$ and $$J_k(c)=\inf\{λ_k(Ω): Ω \textup{quasi-open in}\ \mathbb{R}^m, |Ω|\le 1, \mathcal {P}(Ω)\le c \},$$ where $c>0$, $λ_k(Ω)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(Ω)$, $|Ω|$ denotes the Lebesgue measure of $Ω$, $\mathcal{P}(Ω)$ denotes the perimeter of $Ω$, and where $\mathcal{T}$ is in a suitable class set functions. The latter include for example the perimeter of $Ω$, and the moment of inertia of $Ω$ with respect to its c…
We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $α$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $γ$. We show that for a wide class of density functions the energy admits a minimizer for any value of $γ$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $γ$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the…
We look for the minimizers of the functional $\jla{\la}(\oo)=\la|\oo|-P(\oo)$ among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter $\la$, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.
The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial results to the full range of parameters. We also show that in the regime where the perimeter is dominant, the energy is uniquely minimized by balls.
We consider shape functionals of the form $F_q(Ω)=P(Ω)T^q(Ω)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(Ω)$ denotes the perimeter of $Ω$ and $T(Ω)$ is the torsional rigidity of $Ω$. The minimization and maximization of $F_q(Ω)$ is considered on various classes of admissible domains $Ω$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.
We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition b…
We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. Secondly, we show the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of mini…
The elastic energy of a planar convex body is defined by $E(\Om)=\frac 12\,\int\_{\partial\Om} k^2(s)\,ds$where $k(s)$ is the curvature of the boundary. In this paper we are interested in the minimization problemof $E(\Om)$ with a constraint on the inradius of $\Om$. By contrast with all the other minimization problemsinvolving this elastic energy (with a perimeter, area, diameter or circumradius constraints) for which thesolution is always the disk, we prove here that the solution of this minimization problem is not the disk and we completely characterizeit in terms of elementary functions.
In this paper, we consider the Sub-Laplacian L which consists of sum of squares of smooth vector fields that satisfy Hormander's finite rank condition. We study the Dirichlet problem for this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1) A reversed Holder inequality for the Poisson kernel 2) Harmonic measure (corresponding to L) and surface measure (as well as the H-Perimeter measure) are mutually absolutely continuous 3) A representation (hence solvability of the Dirichlet problem) for solution…
We prove existence of partitions of an open set $Ω$ with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter $s$ is sufficiently close to $1$, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at $120$ degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases.
The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. \par In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a te…
We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions.
In a cylinder $D_T = Ω\times (0,T)$, where $Ω\subset \mathbb{R}^n$, we examine the relation between the $L$-caloric measure, $dω^{(x,t)}$, where $L$ is the heat operator associated with a system of vector fields of Hörmander type, and the measure $dσ_X\times dt$, where $dσ_X$ is the intrinsic $X$-perimeter measure. The latter constitutes the appropriate replacement for the standard surface measure on the boundary and plays a central role in sub-Riemannian geometric measure theory. Under suitable assumptions on the domain $Ω$ we establish the mutual absolute continuity of $dω^{(x,t)}$ and $dσ_X…
Hausdorff Center for Mathematics（豪斯多夫数学中心）；University of Bonn（波恩大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；North Carolina State University（北卡罗来纳州立大学）
We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $L^1+$ (nonlocal) $\operatorname{TV}$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original probl…
Given~$s,σ\in(0,1)$ and a bounded domain~$Ω\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $σ$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(Ω^c)^2) } + \Per_σ\left(\{u>0\},Ω\right), $$ where~$[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and $\Per_σ$ is the fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases…
Maximization and minimization of the principal eigenvalue of the Laplacian with indefinite weight under Dirichlet and Robin boundary conditions on classes of rearrangements
本文研究带不定权重的拉普拉斯算子主特征值在重排类上的最小化与最大化问题，统一处理 Dirichlet 和 Robin 边界条件，并给出 Dirichlet 情形下唯一极大元的完整刻画。
We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $p\to 1$. Tools developed and used in this paper include the inner perimeter measure of a domain.
We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (ϕ(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} Ω$ under Dirichlet boundary conditions, where $Ω\subset {\bf R}^{N}$ is a bounded smooth domain, $ϕ: (0,\infty)\longrightarrow (0,\infty)$ is a suitable continuous function and $f: Ω\times {\bf R} \to {\bf R}$ satisfies the Carathéodory conditions, while $h$ is a measure.
This paper deals with the eigenvalue problem for the operator $L=-Δ-x\cdot \nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $λ_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\in \mathbb{N} $ the following minimization problem $$ \min\left\{λ_k(Ω): \> Ω\>\mbox{quasi-open} \>\mbox{set}, \> \int_Ωe^{|x|^2/2}dx\le c\right\} $$ has a solution.
This paper is concerned with volume-constrained minimization problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on $\mathbb{R}^n$, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrarily to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, …
We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.
We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this inst…
Let $Ω\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-Δ)^s u =λρu$ in $Ω$ with homogeneous Dirichlet boundary condition, where $(-Δ)^s$, $s\in (0,1)$, is the fractional Laplacian operator, $λ\in \mathbb{R}$ and $ ρ\in L^\infty(Ω)$. We study weak* continuity, convexity and Gâteaux differentiability of the map $ρ\mapsto1/λ_1(ρ)$, where $λ_1(ρ)$ is the first positive eigenvalue. Moreover, denoting by $\mathcal{G}(ρ_0)$ the class of rearrangements of $ρ_0$, we prove the existence of a minimizer of $λ_1(ρ)$ when $ρ$ varies…
In this paper we consider a minimization problem of the type $$ I_{β,p}(D;Ω)=\inf\biggl\{\int_Ω\lvert{Dϕ}\rvert^pdx+β\int_{\partial^* Ω}\lvertϕ\rvert^pd\mathcal{H}^{n-1},\; ϕ\in W^{1,p}(Ω),\;ϕ\geq 1 \;\textrm{in}\;D\biggl\}, $$ where $Ω$ is a bounded connected open set in $\mathbb{R}^n$, $D\subset \barΩ$ is a compact set and $β$ is a positive constant. We let the set $D$ vary under prescribed geometrical constraints and $Ω\setminus D$ of fixed thickness, in order to look for the best (or worst) geometry in terms of minimization (or maximization) of $I_{β,p}$. In the planar case, we show that u…
For $Ω_\e=(0,\e)\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \[ \inf_u E^γ_{Ω_\e}(u)\] where \[ E^γ_{Ω_\e}(u):= P_{Ω_\e}(\{u(x)=1\})+γ\int_{Ω_\e}\abs{\nabla{v}}^2\,dx \] and the minimization is taken over competitors $u\in BV(Ω_\e;\{\pm 1\})$ satisfying a mass constraint $\fint_{Ω_\e}u=m$ for some $m\in (-1,1)$. Here $P_{Ω_\e}(\{u(x)=1\})$ denotes the perimeter of the set $\{u(x)=1\}$ in $Ω_\e$, $\fint$ denotes the integral average and $v$ denotes the solution to the Poisson problem \[ -Δv=u-m\;\mbox{in}\;Ω_\e,\quad\nab…
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^β$ that provides the existence of a minimizer for any volume constraint, and we study the geometry of big minimizers. Then we provide a numerical method to adress this variational problem.
In this paper, we deal with an obstacle placement problem inside a disk, that can be formulated as an energy minimization problem with respect to the rotations of the obstacle about its center, with respect to the translations of the obstacle within a planar disk and also with respect to the expansion or contraction of the obstacle inside the disk. We also include its sensitivity with respect to the boundary data. We consider an inhomogeneous Dirichlet Boundary Value Problem on $ B \setminus P$ as follows: $Δu =0$ on $ B \setminus P$, $u=0$ on $\partial P$, $u=M$, a nonzero constant on $\parti…
We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $Ω$}, & U|_{\partialΩ}&=ϕ. \end{eqnarray*} Here $Ω$ a smooth bounded domain in $R^n$, $f: R^N\to R$ is a smooth function, $U:Ω\to R^N$ is the unknown vector-valued function, $ϕ:\barΩ\to R^N$ is a given vector-valued $C^2$ function, $f'$ is the gradient of the function $f$ with respect to the variable $U$. Such problems arise in population dynamics and Differential Geometry. The difficulty of studying this problem is that this…
We consider the energy modeling a two component Bose-Einstein condensate in the limit of strong coupling and strong segregation. We prove the $Γ$-convergence to a perimeter minimization problem, with a weight given by the density of the condensate. In the case of equal mass for the two components, this leads to symmetry breaking for the ground state. The proof relies on a new formulation of the problem in terms of the total density and spin functions, which turns the energy into the sum of two weighted Cahn-Hilliard energies. Then, we use techniques coming from geometric measure theory to cons…
In his deep and prolific investigations of heat diffusion, Lamé was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular he derived explicit results for the Dirichlet and Neumann cases using an ingenious change of variables. The relevant eigenfunctions are complicated infinite series in terms of his variables. Here we first show that boundary value problems with simple boundary conditions, such as the Dirichlet and the Neumann problems, can be solved in an elementary manner. In particular for these problems, the unknow…
In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $Ω$. Here with $N$-Cluster we mean a family of $N$ sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any $N$-cluster attaining such a minimum a Cheeger $N$-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger $N$-clusters in a general ambient space dimension and we give …
We consider a class of generalized antiferromagnetic local/nonlocal interaction functionals in general dimension, where a short range attractive term of perimeter type competes with a long range repulsive term characterized by a reflection positive power law kernel. Breaking of symmetry with respect to coordinate permutations and pattern formation for functionals in this class have been shown in~\cite{gr,dr_arma} and previously by~\cite{gs_cmp} in the discrete setting, for a smaller range of exponents. Global minimizers of such functionals have been proved in~\cite{dr_arma} to be given by peri…
This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as $\min\Big\{{g}(F_1(Ω_1),\dots,F_h(Ω_h))+ m\vert\,\bigcup_{i=1}^hΩ_i\vert :\ Ω_i\subset D,\ Ω_i\cap Ω_j =\emptyset\Big\},$ where $D\subset\mathcal{R}^d$ is a given bounded open set, $\vertΩ_i\vert$ is the Lebesgue measure of $Ω_i$ and $m$ is a positive constant. For a large class of such functionals, we analyse qualitative properties of the cells $Ω_i$ and the interaction between them. Each cell is itself subsolution for a (single phase) shape optimization problem, from which we ded…
