We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter $η$ to represent the size of the domains of the minority phase, and study the resulting droplet regime as $η\to 0$. By considering confinement densities which are spatially variable and attain a nondegenerate maximum, we present a two-stage asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter $M$ which gives the total volume of the droplets at order $η^3$ and its relation to existence and non-existence of Gamow's Liquid Drop model on $\mathbb{R}^3$. For large values of $M$, the minority phase splits into several droplets at an intermediate scale $η^{1/3}$, while for small $M$ minimizers form a single droplet converging to the maximum of the confinement density.
Young's law for a nonlocal isoperimetric model of charged capillarity droplets
CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）；Department of Mathematics, University of Pisa（比萨大学）；CMAP, École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）
We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.
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.
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 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…
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 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…
A class of nonlocal hypoelliptic operators and their extensions
在本文中，我们研究的非局部方程的分数权力的hypoellipticity运营商的形式$$\mathscr K u = \mathscr A u - \partial_t u \overset{def}{=} \operatorname{tr}(Q \nabla^2 u) + <BX,\nabla u> - \partial_t u,$$介绍了Hörmander在他的1967年hypoellipticity文件。证明了非局部算子$(-\mathscr K)^s$和$(-\mathscr A)^s$可以实现为双退化扩张问题的Dirichlet-to-Neumann映射.我们解决了这样的问题，在$L^\infty$，并在$L^p$为$1\leq p<\infty$时，$\operatorname{tr}(B)\geq 0$。在以后的工作中，我们使用这种微积分建立了一些新的Sobolev不等式和等周不等式。
This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove the boundary of the droplet is a chord-arc curve with its normal vector field in the VMO space. In fact, the boundary curves of such droplets belong to the so-called Weil-Petersson class. In addition, the asymptotic behavior of the optimal shape when t…
Motivated by Gamow's liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by $P(E)-γP_\varepsilon(E)$, with $0<γ<1$ and $P_\varepsilon$ a nonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes. We prove that unit area minimizers are disks for $\varepsilon>0$ small enough. More precisely, we first show that in dimension $2$, minimizers are necessarily convex, provided that $\varepsilon$ is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their bound…
