Existence of minimizers of functionals involving the fractional gradient in the abscence of compactness, symmetry and monotonicity
Hichem Hajaiej
我们建立了一般的假设，在此假设下，一个包含分数阶梯度和局部非线性的约束变分问题允许极小。
We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.
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Some monotonicity results for minimizers in the calculus of variations
我们得到了$$ \int F(\nabla u, u, x) dx $$型一般能量泛函的极小值和稳定解的单调性性质的假设下，一定的积分最多二次增长在无穷远。作为结果，我们得到了几个刚性的整体解的低维结果。
We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in planar domains. Our formula follows the spirit of Bucur-Luckhaus, but works with the David-Léger entropy instead of the energy. Interestingly, this allows for a sharp truncation threshold. In particular, our monotonicity formula is able to discriminate between points at a $C^1$ interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.
A rigidity result for global Mumford-Shah minimizers in dimension three
本文研究了$\R^N$中的全局Mumford-Shah极小元，该极小元是Bonnet作为Mumford-Shah极小元的爆破极限引入的.当奇异集$K$包含在一个足够光滑的锥中时，我们证明了一个新的关于$u$能量的单调性公式.然后，我们使用这种单调性证明，对于任何减少全球极小$(u,K)$在$\R^3$，如果$K$包含在一个半平面和接触它的边缘，那么它是半平面本身。这部分回答了盖伊·大卫的一个问题。
Optimization of the principal eigenvalue of the Neumann Laplacian with indefinite weight and monotonicity of minimizers in cylinders
本文研究了带不定权重的 Neumann Laplacian 主特征值优化问题，证明了极小化子的存在性与特征刻画，并揭示了柱体区域中极小化子沿母线方向的单调性。
Minimizers of a class of constrained vectorial variational problems: Part I
本文证明了一类多约束变分问题极小解的存在性。我们考虑系统涉及的非线性，不满足紧凑性，单调性，既不对称性。我们的方法依赖于集中-紧性方法。在第二部分中，我们将用密度矩阵方法处理另一类被积函数的正交约束问题。
Regularity for almost minimizers with free boundary
本文研究了泛函\begin{equation*} J(u)=\int_Ω|\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x) \end{equation*}的几乎极小元的局部正则性，其中$q_\pm \in L^\infty(Ω)$ .几乎极小化器不像极小化器那样满足PDE或单调性公式（参见\cite{AC}、\cite{ACF}、\cite{CJK}、\cite{W}）。然而，我们成功地证明了他们是局部Lipschitz，这是最佳的正则极小。
We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter go…
Weiss monotonicity and capillary hypersurfaces
本文证明当毛细角趋于零时，毛细面积密度收敛到 Weiss 能量密度，并据此获得与角度无关的曲率估计和毛细极小元正则性结果。
We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighborhood of its minimum. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential.
We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.
Free boundary regularity for almost minimizers of the parabolic Signorini problem
本文针对抛物型Signorini问题中的几乎极小化子，通过建立Weiss型单调性公式和Almgren型频率公式，结合epiperimetric不等式，证明了正则自由边界点的优化增长及正则集的正则性。
We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the $C^{1,γ}$ regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperim…
Given a bounded $C^2$ domain $Ω\subset{\mathbb R}^d$ with $d\geq3$, we prove a sharp inequality which relates the perimeter of ${\partialΩ}$ to the endpoint Gagliardo seminorm in $W^{r,2}({\partialΩ})$, corresponding to $r=0$, of the normal vector field on ${\partialΩ}$. The proof of the inequality relies on the use of Bessel potentials and a monotonicity formula; we also show that balls are the unique minimizers. For $1/2<r<1$, the Gagliardo seminorm of the normal vector field on ${\partialΩ}$ is related to a fractional second fundamental form which arises in the study of nonlocal perimeters …
We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional $$ϕ_p(r,u,x_0)=\frac1{r^4}\int_{B_r(x_0)}\frac{|\nabla u^+(x)|^p}{|x-x_0|^{N-2}}dx\int_{B_r(x_0)}\frac{|\nabla u^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\in\partial\{u>0\}$ and $u$ is a minimizer of the functional $$J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le …
In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in L^\infty(\mathcal O)$. Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a non-degeneracy assumption on $q_\pm$, the free boundary is uniformly rectifiable. Furthermore, when $q_-\equiv 0$, and $q_+$ is Hölder continuous we s…
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…
We study an intrinsic model for collective behaviour on the hyperbolic space $\bbh^\dm$. We investigate the equilibria of the aggregation equation (or equivalently, the critical points of the associated interaction energy) for interaction potentials that include Newtonian repulsion. By using the method of moving planes, we establish the radial symmetry and the monotonicity of equilibria supported on geodesic balls of $\bbh^\dm$. We find several explicit forms of equilibria and show that one such equilibrium is a global energy minimizer. We also consider more general potentials and utilize a te…
We consider a fractional version of Gauss capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional minimizing cone, the one determined by the fractional version of Young's law.
A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler-Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.
The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions $n…
We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the…
Continuity and monotonicity of weighted perimeters of convex bodies
该研究针对$\boldsymbol{\text{R}}^n$中凸体的加权周长，证明其关于Hausdorff度量的连续性、特定径向权下包含关系的单调性，将结果用于证明形状优化问题极小元的存在性。
In this work, we prove an existence result for an optimal partition problem of the form $$\min \{F_s(A_1,\dots,A_m)\colon A_i \in \mathcal{A}_s, \, A_i\cap A_j =\emptyset \mbox{ for } i\neq j\},$$ where $F_s$ is a cost functional with suitable assumptions of monotonicity and lowersemicontinuity, $\mathcal{A}_s$ is the class of admissible domains and the condition $A_i\cap A_j =\emptyset$ is understood in the sense of the Gagliardo $s$-capacity, where $0<s<1$. Examples of this type of problem are related to the fractional eigenvalues. In addition, we prove some type of convergence of the $s$-mi…
This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Niren…
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…
On a class of optimal partition problems related to the Fuč\'ık spectrum and to the monotonicity formulae
在本文中，我们给出了一个统一的方法来解决非线性分析的不同领域中出现的一些问题，即：（a）的结构的研究的Fuelik\'ık频谱和（b）可能的变体和扩展的单调性公式的Alt-Caffarelli-Friedman \cite{acf}。本文的第一部分给出了一类涉及Laplace算子第一特征值的最优划分问题。除了建立最优划分的存在性，我们还发展了极值条件和极小值的正则性理论。作为这种方法的第一个应用程序，我们给出了一个新的变分表征的第一曲线的Fuelik\'ık谱的拉普拉斯算子，迅速适应更一般的运营商。第二部分证明了多个次调和分量情形下的一个单调性公式，并推广到一类反应扩散方程的解，给出了几个Liouville型定理。
Minimization of a fractional perimeter-Dirichlet integral functional
我们考虑一个最小化问题，结合Dirichlet能量与非局部周长的水平集，即$$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$与$σ\in(0,1)$。我们得到的最小和他们的自由边界$\p \{u>0\}$使用爆破分析的正则性结果。我们还将给出密度估计、单调性公式、Euler-Lagrange方程和延拓问题的相关结果。
We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.
One-dimensional Neel walls under applied external fields
我们提出了一个详细的分析一维Néel壁在薄的单轴铁磁薄膜中的存在下的面内施加的外场在垂直于易磁化轴的方向。在简化的一维薄膜模型中，我们提出了一个非局部的变分问题，其极小值由一维Néel壁廓线给出。我们证明存在性，唯一性（平移和反射），正则性，严格单调性和精确渐近的衰减的极小考虑变分问题。
Monotonicity formulas in positively curved settings with applications to two-phase free boundary problems
本文针对二维正曲率环境构造ACF泛函的变体并证明其几乎单调性公式，借此得到两相自由边界问题极小元的新利普希茨界，常数仅依赖总积分曲率，还恢复了特定情形下的已有利普希茨界。
On a class of coupled obstacle systems
本文针对新型合作与竞争型耦合障碍系统，用变分法约束极小值点证明解的存在性并得到最优正则性，还借助爆破、Weiss型与Monneau型单调公式研究自由边界正则性，为自由边界问题研究提供了新支撑。
Given~$s,σ\in(0,1)$ and a bounded domain~$Ω\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $σ$-perimeter type $$ [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…
This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in $\mathbb{R}^n$ which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is…
Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian
在$\re^{2m}$空间中建立了椭圆型分数阶方程$(-Δ)^{1/2}u=f(u)$鞍形解的存在性和定性性质，其中$f$是双稳态型的.这些解对于西蒙斯锥是奇数，对于每个坐标是偶数。更准确地说，我们证明了鞍形解的存在性，在每个偶数维$2m$，以及它的单调性，渐近行为，和不稳定的尺寸$2m=4$和$2m=6$。这些结果是相关的模拟分数方程的猜想德乔治的1-D对称性的某些解决方案。鞍形的解决方案是最简单的候选人，除了1-D的解决方案，是全球最小的高维，尚未建立的属性。
Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring
本文研究三维单位球上受切向约束的$\mathbb{S}^2$值极小调和映射，获得尊重切向性的单调性公式以证明边界正则性，并给出极小元具有对称性的新充分条件，最终刻画其恰有两个位于边界相对点的点奇点。
Periodic waves in the fractional Korteweg-de Vries equation have been previously characterized as constrained minimizers of energy subject to fixed momentum and mass. Here we characterize these periodic waves as constrained minimizers of the quadratic form of energy subject to fixed cubic part of energy and the zero mean. This new variational characterization allows us to unfold the existence region of travelling periodic waves and to give a sharp criterion for spectral stability of periodic waves with respect to perturbations of the same period. The sharp stability criterion is given by the m…
We prove partial regularity of stationary solutions and minimizers $u$ from a set $Ω\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_ΩF(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an $ε$-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.
For open radial sets $Ω\subset \mathbb{R}^N$, $N\geq 2$ we consider the nonlinear problem \[ (P)\quad Iu=f(|x|,u) \quad\text{in $Ω$,}\quad u\equiv 0\quad \text{on $\mathbb{R}^N\setminus Ω$ and }\lim_{|x|\to\infty} u(x)=0, \] where $I$ is a nonlocal operator and $f$ is a nonlinearity. Under mild symmetry and monotonicity assumptions on $I$, $f$ and $Ω$ we show that any continuous bounded solution of $(P)$ is axial symmetric once it satisfies a simple reflection inequality with respect to a hyperplane. In the special case where $f$ does not depend on $|x|$, we show that any nonnegative nontrivia…
We consider the infinite horizon risk-sensitive problem for nondegenerate diffusions with a compact action space, and controlled through the drift. We only impose a structural assumption on the running cost function, namely near-monotonicity, and show that there always exists a solution to the risk-sensitive Hamilton-Jacobi-Bellman (HJB) equation, and that any minimizer in the Hamiltonian is optimal in the class of stationary Markov controls. Under the additional hypothesis that the coefficients of the diffusion are bounded, and satisfy a condition that limits (even though it still allows) tra…
This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{λ_1^s(Ω)+\cdots+λ_m^s(Ω) + Λ\mathcal{L}_n(Ω)\colon Ω\subset D \mbox{ s-quasi-open}\right\} $$ where $Λ>0, D\subset \mathbb{R}^n$ is a bounded open set and $λ_i^s(Ω)$ is the $i$-th eigenvalues of the fractional Laplacian on $Ω$ with Dirichlet boundary condition on $\mathbb{R}^n\setminus Ω$. We first prove that the first $m$ eigenfunctions on an optimal set are locally Hölder continuous in the class $C^{0,s}$ and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss …
Uniform estimates for positive solutions of semilinear elliptic equations and related Liouville and one-dimensional symmetry results
考虑光滑区域上可能无界的半线性椭圆型方程的Dirichlet边界条件.在适当的假设下，我们推导出一个条件的大小域，这意味着存在一个正解满足一致逐点估计。这里，均匀意味着估计与域无关。我们的方法的主要优点是，它允许我们删除一个限制性的单调性假设，这是强加在最近的文件。此外，我们可以删除一个非简并条件，在后者的参考。进一步推广了一个关于半线性椭圆非线性本征值问题的结果。此外，我们还研究了相应的奇异扰动问题的全局极小解的边界层。对于上述应用程序，我们的方法是基于一个结果的细化，关于大球的相关能量的全局极小的行为，受狄利克雷条件。结合整体分歧理论和滑动方法，我们可以证明解远离其节点集的一致估计。结合我们的方法与先验估计，我们得到的blow-up，加倍引理，和已知的Liouville型定理，我们可以给出一个新的证明，一个已知的Liouville型定理不使用边界blow-up解决方案。我们也可以提供一个替代的证明，和一个有用的推广，刘维尔定理，涉及存在的障碍。利用后一种推广，我们考虑了具有混合边界条件的奇异扰动问题。此外，我们证明了Allen-Cahn型方程的某些整体解的一些新的一维对称性和刚性性质，以及在半空间，凸圆柱区域。特别地，我们在二维空间中给出了Gibbons猜想的一个新的证明。
Lectures on Existence and Regularity of Anisotropic Minimal Surfaces
本迷你课程概述各向异性极小曲面的存在性与正则性理论进展，研究各向异性普拉托问题解、临界点正则性，并将结果应用于闭黎曼流形中最优正则各向异性极小超曲面的构造。
We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this pa…
Morrey estimates for the gradient in non-linear variational transmission problems
本文研究非线性变分传输问题中梯度的Morrey正则性，通过能量衰减估计证明梯度在Morrey空间中的局部可积性。
A new variational approach to the stability of gravitational systems
我们考虑三维引力VlasovPoisson系统，它描述了恒星系统在自身引力作用下的力学状态。天体物理学中的一个著名猜想是，作为其微观能量的非增函数的稳态解在流动的作用下是非线性稳定的。这是证明了在线性水平的几个作者的基础上的开创性工作安东诺夫在1961年。从那时起，标准变分技术的基础上的浓度紧性方法介绍了P. - L. Lions在1983年得到了基态型定态解子类的非线性稳定性。本文受物理学文献（Lynden-Bell 94，Wiechen-Ziegler-Schindler MNRAS 88，Aly MNRAS 89）的启发，利用广义对称重排变换下哈密顿量的单调性，证明了在等测度约束下非增稳定解是哈密顿量的局部极小，并从适当的极小序列中提取了紧性.这意味着在径向对称扰动下非增各向异性定态的非线性稳定性。
Existence and nonexistence of normalized solutions for nonlinear Schrödinger equation involving combined nonlinearities in bounded domain
本文研究了非线性Schrödinger方程在有界域中混合非线性的解的存在性和不存在性，通过能量极小值、单调性技巧、移动平面法等方法，探讨了不同条件下解的性质。
Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality
本文利用修改后的 Simon 的 Li-Yau 不等式和几何测度论工具，证明了初始能量低于特定阈值的带 Dirichlet 边界数据的圆柱旋转曲面 Willmore 流的全局存在性，并改进了平稳问题中 Willmore 极小化子的存在性结果。
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