Davide Carazzato, Nicola Fusco, Aldo Pratelli
We consider functionals given by the sum of the perimeter and the double integral of some kernel $g:\mathbb R^N\times\mathbb R^N\to \mathbb R^+$, multiplied by a "mass parameter" $\varepsilon$. We show that, whenever $g$ is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as $\varepsilon\ll 1$.
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We deduce that charged liquid droplets minimizing Debye-Hückel-type free energy are spherical in the small charge regime. The variational model was proposed by Muratov and Novaga in 2016 to avoid the ill-posedness of the classical one. By combining a recent (partial) regularity result with Selection Principle of Cicalese and Leonardi, we prove that the ball is the unique minimizer in the small charge regime.
Stability and minimality of the ball for attractive-repulsive energies with perimeter penalization
本文研究带周长惩罚的吸引-排斥能量中球的稳定性与极小性，证明参数空间中存在分离稳定与极小区域的曲线，并发现小扰动下球稳定区域至少有两个不连通部分。
We study the existence, uniqueness and minimality of critical points of the form $m_{\varepsilon,η}(x) = (f_{\varepsilon,η}(|x|)\frac{x}{|x|}, g_{\varepsilon,η}(|x|))$ of the functional \[ E_{\varepsilon,η}[m] = \int_{B^N} \Big[\frac{1}{2} |\nabla m|^2 + \frac{1}{2\varepsilon^2} (1 - |m|^2)^2 + \frac{1}{2η^2} m_{N+1}^2\Big]\,dx \] for $m=(m_1, \dots, m_N, m_{N+1}) \in H^1(B^N,\mathbb{R}^{N+1})$ with $m(x) = (x,0)$ on $\partial B^N$. We establish a necessary and sufficient condition on the dimension $N$ and the parameters $\varepsilon$ and $η$ for the existence of an escaping vortex solution $(…
Global minimality of the degree-one Ginzburg-Landau vortex solution in the unit ball in dimensions $n \geq 2$
本文证明了在单位球内，对于任意Ginzburg-Landau参数，旋转对称涡旋解是唯一的全局极小值，维度$n\geq 2$。
Local and global minimality results for a nonlocal isoperimetric problem on R^N
考虑定义在全空间$\R^N$中的非局部等周问题，其非局部部分由指数为$α\in(0, N-1)$的里斯位势给出.我们证明了具有正二阶变分的临界组态是局部极小元，并且满足关于$L^1$ -范数的一个定量不等式。这个标准提供了一个（明确确定的）临界阈值的存在，确定球是一个局部极小的体积的间隔，并允许解决几个全球极小问题。
Minimality of vortex solutions to Ginzburg--Landau type systems for gradient fields in the unit ball in dimension $N\geq 4$
本文证明了在维数 $N\geq 4$ 的单位球中，度数为 1 的涡旋解是梯度场 Ginzburg--Landau 泛函（Aviles--Giga 模型）的唯一极小化子，并提出了两种证明方法。
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…
The minimality of the map x/|x| for weighted energy
本文研究了$E\_{p,f}= \int\_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$型加权能量泛函从欧氏单位球$\mathbf{B}^n$到其边界$\mathbb{S}^{n-1}$的映射$\frac{x}{\|x\|}$的极小性，其中$f$是非负函数.我们证明了在以下两种情况下：i）$p=1$和$f$是非减的，i））$p$是一个整数，$p \leq n-1$和$f= r^α$与$α\geq 0$，映射$\frac{x}{\|x\|}$在$W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$中与$\frac{x}{\|x\|}$在$\partial \mathbf{B}^n$上重合的映射中最小化$E\_{p,f}$.我们还研究了$ f(r)= r^α$与$-n+2 < α< 0$的情况，并证明了$\frac{x}{\|x\|}$不会使$α$接近$-n+2$的$E\_{p,f}$最小化，当$n \geq 6$时，$α$接近$4-n$。
On minimizers in the liquid drop model
本文证明了在伽莫夫液滴模型中，体积不超过1的圆球是唯一的极小化子。
An Induction Principle for the Weighted p-Energy Minimality of x/|x|
本文研究了从欧氏单位球$\mathbf{B}^{n}$到其边界$\mathbb{S}^{n-1}$的映射$x/\|x\|$对加权能量泛函$E^n\_{p,α}(u)=\int\_{\mathbf{B}^{n}} \|x\|^α\|\nabla u\|^p dx$的极小化性质.我们建立了如下的归纳原理：如果映射$\frac{x}{\|x\|}:\mathbf{B}^{n+1}\to \mathbb{S}^n$在满足$u(x)=x$ on $\mathbb{S}^n$的映射$u: \mathbf{B}^{n+1}\to \mathbb{S}^n$中最小化$E^{n+1}\_{p,α}$，则映射$\frac{y}{\|y\|}:\mathbf{B}^n\to\mathbb{S}^{n-1}$在满足$v(y)=y$ on $\mathbb{S}^{n-1}$的映射$v: \mathbf{B}^n\to\mathbb{S}^{n-1}$中最小化$E^{n}\_{p,α+1}$.这一结果使我们能够扩大$x/\|x\|$使$E^n\_{p,α}$最小化的$p$和$α$的值的范围。
We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,γ}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particul…
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…
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, …
Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case
本文证明当1<p<2时，最佳分数阶Sobolev常数的体积约束局部极小化区域必为球，通过分数阶Hadamard公式、加权奇异窄域极大值原理和有限边界展开克服移动平面法障碍，覆盖完整参数范围。
We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minim…
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…
Uniform $C^{1,α}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter
本文研究周长非局部扰动泛函的几乎极小元正则性，建立了参数ε趋于0时一致的C¹,ᵅ正则性判据，证明小ε下体积约束极小元为球，适用于强非局部相互作用的Gamow型大质量问题。
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