Regularity and relaxed problems of minimizing biharmonic maps into spheres

Minimizing under relaxed symmetry constraints: Triple and $N$-junctions

We consider a nonnegative potential $W:\mathbb{R}^2\rightarrow\mathbb{R}$ invariant under the action of the rotation group $C_N$ of the regular polygon with $N$ sides, $N\geq 3$. We assume that $W$ has $N$ nondegenerate zeros and prove the existence of a $N$-junction solution to the vector Allen-Cahn equation. The proof is variational and is based on sharp lower and upper bounds for the energy and on a new pointwise estimate for vector minimizers.

带边界距离函数的尖锐二阶不等式及其在p-双调和奇异问题中的应用

Sharp second order inequalities with distance function to the boundary and applications to a p-Biharmonic singular problem

本文将带边界距离函数奇异项的Hardy-Rellich不等式推广到L^p情形，给出尖锐常数或新界，并应用于p-双调和奇异问题解的存在性与非存在性研究。

A note on $n$-axially symmetric harmonic maps from $B^3$ to $S^2$ minimizing the relaxed energy

For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Hardt, Lin and Poon for the case n=1.

向量值、高阶及非极小化Bernoulli问题的最优正则性

Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems

本文提出不依赖粘性解或比较方法的新技术，证明向量值、高阶及非极小化Bernoulli自由边界问题解的Lipschitz最优正则性，并推广至弹性、双调和及Stokes方程及二维标准问题。

双调和Brézis-Nirenberg问题的能量渐进行为与爆炸现象

Energy asymptotics and blow-up phenomena for biharmonic Brézis-Nirenberg problem

本文研究了n≥8维下双调和Brézis-Nirenberg问题在Navier边界条件下的商泛函，通过匹配上界和下界估计，建立了能量差S(0)-S(εV)的精确渐进行为，并描述了近极小序列的爆炸剖面及集中点的性质。

On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions

In this paper we estimate from above the area of the graph of a singular map $u$ taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2-$ embedded curves that meet transversely at only one point of the disk. We show that the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to "fill the hole" in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces,…

测度在Wasserstein距离下的一维逼近

One-dimensional approximation of measures in Wasserstein distances

本文提出一种变分方法，用一维集合上的均匀测度逼近原测度，通过最小化Wasserstein距离并引入支撑长度正则化，提出松弛形式证明解的存在性，并证明其在$\mathbb{R}^2$中与原问题等价且最优解具有Ahlfors正则性。

Relaxed solutions for incompressible inviscid flows: A variational and gravitational approximation to the initial value problem

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation which generates a set of "relaxed", "multi-stream", version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an atte…

Relaxed multi-marginal costs and quantization effects

We propose a duality theory for multi-marginal repulsive cost that appear in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the $N$-marginals relaxed cost in terms of a stratification formula which takes into account all $k$ interactions with $k\le N$. We then develop a duality framework involving continuous functions vanishing at infinity and deduce prim…

On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity

In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain of the plane with values in the plane and discontinuous on a regular simple curve with two endpoints. We show that, under suitable assumptions, the relaxed area does not exceed the area of the regular part of the map, with the addition of a singular term measuring the area of a disk type solution of the Plateau's problem spanning the two traces of the map on the jump. The result is valid also when the area minimizing surface has self intersections. A key element in…

非退化梯度Young测度与非线性薄膜的维数约化

Nondegenerate gradient Young measures and dimension reduction for nonlinear membranes

本文从三维非线性弹性严格推导薄膜理论，引入保持取向约束与惩罚体积消失的能量增长，通过非退化梯度Young测度获得Γ-极限，唯一确定捕捉微结构的非松弛膜能量密度，并恢复Le Dret-Raoult经典松弛能量，利用分段等距构造解决生成测度的关键困难。

含液相等周簇

Isoperimetric clusters with a liquid phase

本文研究湿泡沫的等周问题，提出松弛能量处理液体腔室坍缩为零厚度薄膜的情况，证明极小化序列极限的存在性与正则性，并揭示液体体积消失时能量收敛于干泡沫。

调和偶极子与三维弹性中 neo-Hookean 能量的松弛

Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity

针对三维 neo-Hookean 能量最小化中无空化映射空间因偶极子导致的非紧性问题，本文在轴对称映射框架下提出最小化空间与显式惩罚能量，并揭示其与调和映射松弛能量的联系。

弹性薄片浮于液滴状液体基底上的褶皱

Wrinkling of an elastic sheet floating on a liquid sphere

研究弹性薄片在液滴状液体基底上的褶皱现象，通过变分方法分析能量最小化，推导厚度比展开式并建立上下界。