Global solutions to special Lagrangian equations

A Bernstein problem for special Lagrangian equations

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

Singular Solution to Special Lagrangian Equations

We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.

特殊拉格朗日方程的常秩定理

A constant rank theorem for special Lagrangian equations

本文证明了特殊拉格朗日方程和二次 Hessian 方程鞍点解的常秩定理，并由此得到次临界相位下特殊拉格朗日方程的 Liouville 型刚性结果。

A Bernstein type result for special Lagrangian submanifolds

Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.

特殊拉格朗日曲率方程的先验内部估计

A priori interior estimates for special Lagrangian curvature equations

本文针对特殊拉格朗日曲率方程，在临界相和凸情形下建立了先验内部曲率估计，并证明了任意常数相的先验内部梯度估计。

Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.

Minimal surface systems, maximal surface systems, and special Lagrangian equations

We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.

具有多个孤立奇点的特殊拉格朗日量

Special Lagrangians with multiple isolated singularities

该研究将Caffarelli-Hardt-Simon扰动论证推广到特殊拉格朗日量领域，证明正则特殊拉格朗日锥的桥接原理，进而得到具有多个给定孤立锥形奇点的特殊拉格朗日子流形的存在性。

On the second boundary value problem for special Lagrangian curvature potential equation

This is a sequel to [1] and [2], which study the second boundary problem for special Lagrangian curvature potential equation. As consequences, we obtain the existence and uniqueness of the smooth uniformly convex solution by the method of continuity through carrying out a-priori estimate on the solutions.

Moduli space and deformations of special Lagrangian submanifolds with edge singularities

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to study the moduli space of deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

Deformations of Asymptotically Conical Special Lagrangian Submanifolds

McLean studied the deformations of compact special Lagrangian submanifolds, showing in particular that they come in moduli spaces whose dimension depends only on the topology of the submanifold. In this article we study the analogous problem for non-compact, "asymptotically conical" SL submanifolds, with respect to various "boundary conditions at infinity".

特殊拉格朗日方程的半凸粘性解

Semi-convex viscosity solutions of the special Lagrangian equation

本文研究特殊拉格朗日方程在近负相位、小半凸条件下的粘性解，证明其光滑性、内部导数与指数型黑塞估计，并给出尖锐性反例及次临界相位整解的新刘维尔定理。

三维特殊拉格朗日方程的非凸次水平集

Nonconvex Sublevel Sets for the Three-Dimensional Special Lagrangian Equation

针对三维特殊拉格朗日方程，在相位Θ∈(π/2,π)时构造特定凸区域，发现其零狄利克雷解存在非凸负次水平集。

Regularizing a singular special Lagrangian variety

Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle condition (which always holds in dimension $n=3$). Then, $M_{1} \cup M_{2}$ is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds $M_α$ with boundary that converges to $M_{1} \cup M_{2}$ in a suitable topology. T…

特殊拉格朗日型方程的Neumann问题

The Neumann problem of special Lagrangian type equations

该研究针对带临界与超临界相位的特殊拉格朗日型方程Neumann问题，通过建立二阶一致先验估计、直接证明边界双法向导数估计等技术，结合连续性方法得到存在性结果，还解决了经典Neumann问题。

通过倍增法得到特殊拉格朗日方程的Hessian估计

Hessian estimates for special Lagrangian equation by doubling

本文通过倍增法给出了特殊拉格朗日方程内部Hessian估计的新证明，不依赖Michael-Simon平均值不等式，扩展了Korevaar对极小超曲面梯度估计的点wise证明。

Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz

The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.

带约束的一般相位上特殊拉格朗日方程的 Hessian 估计的加倍论证

Doubling Argument of the Hessian Estimate for the Special Lagrangian Equation on General Phases with Constraints

本文建立加倍论证方法，在不依赖 Michael-Simon 均值不等式的情况下，得到带一般相位约束的特殊拉格朗日方程的 Hessian 估计，并建立了具独立意义的 Alexandrov 型定理。

特殊Lagrange方程的Liouville型定理与Hessian估计

Liouville type theorems and Hessian estimates for special Lagrangian equations

本文针对特殊Lagrange方程在特定“凸性”条件下证明了Liouville型定理，核心方法是通过Neumann-Poincaré不等式和均值不等式建立解的整体Hessian估计，并进一步推导了该条件及超临界相位下的内部Hessian估计。

Special Lagrangian cones with higher genus links

For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.

A one-sentence proof of the Bernstein type theorem for special Lagrangian equation in two dimensions

The aim of this expository article is to reveal the equivalence of Jörgens' Theorem on the two dimensional unimodular Hessian equation and Fu's Theorem on the two dimensional special Lagrangian equation.

特殊拉格朗日方程半凸粘性解的几何正则性

Geometric regularity for semi-convex viscosity solutions to the special Lagrangian equation

本文证明三维次临界特殊拉格朗日方程半凸粘性解梯度图面积最小且光滑，并推广至更高维度，推进了Wang-Yuan猜想。

特殊拉格朗日方程在无穷远处的解

Solutions of the Special Lagrangian Equation near Infinity

本文研究特殊拉格朗日方程在无穷远处的渐近行为，借助修正的 Kelvin 变换以原点附近单个函数刻画余项，并给出其在奇偶维数下的正则性差异。

哈密顿稳态与特殊拉格朗日图的尖锐部分正则性

Sharp partial regularity of Hamiltonian stationary and special Lagrangian graphs

该研究证明了任意光滑近凯勒流形中哈密顿稳态拉格朗日 Lipschitz 子流形的弱解在豪斯多夫维数至多为 $n-5$ 的奇异集外光滑，并构造反例说明该维数估计对特殊拉格朗日图是尖锐的。

On the exterior Dirichlet problem for special Lagrangian equations

In this paper, we establish the existence and uniqueness theorem of the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity.

正拉格朗日子流形的测地线来自具有边界的特殊拉格朗日子流形

Geodesics of positive Lagrangians from special Lagrangians with boundary

研究正拉格朗日子流形空间中的测地线，通过柱形变换揭示其与特殊拉格朗日柱体的关系，并证明相关空间的开性，提供非对称维度下的测地线解例。

特殊拉格朗日方程的非$C^1$解

Non $C^1$ solutions to the special Lagrangian equation

本研究构造了特殊拉格朗日方程的Lipschitz连续但不具备$C^1$光滑性的粘性解，丰富了该方程解的正则性相关研究结果。

The degenerate special Lagrangian equation on Riemannian manifolds

We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Ri…

具有定量精确动力学的拉格朗日平均曲率流的有限时间奇点

Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

构建几乎校准的拉格朗日平均曲率流，从接近特殊拉格朗日的初值出发，在有限时间产生II型奇点，给出曲率爆破率，奇点切流为特殊拉格朗日锥，爆破极限是光滑特殊拉格朗日去奇异化。

On the Dirichlet problem for special Lagrangian curvature potential equation

In this paper, we study a class of special Lagrangian curvature potential equations and obtain the existence of smooth solutions for Dirichlet problem. The existence result is based on a priori estimates of global $C^{0}$, $C^{1}$ and $C^{2}$ norms of solutions under the assumption of existence of a subsolution.

Asymptotic expansion at infinity of solutions of special Lagrangian equations

We obtain a quantitative high order expansion at infinity of solutions for a family of fully nonlinear elliptic equations on exterior domain, refine the study of the asymptotic behavior of the Monge-Ampère equation, the special Lagrangian equation and other elliptic equations, and give the precise gap between exterior maximal (or minimal) gradient graph and the entire case.

The degenerate special Lagrangian equation

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in $\mathbb{C}^n.$ Existence of geodesics in the space of positive Lagrangians is an important step in a program for proving existence and uniqueness of special Lagrangians. Moreover, it would imply certain cases of the strong Arnold conjecture from Hamiltonian dynamics. We show the DSL is degenerate elliptic. We introduce a space-time Lagrangian angle for one-parameter families …

A Bernstein problem for special Lagrangian equations in exterior domains

We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.

次临界特殊拉格朗日方程的内部Hölder与梯度估计失效

Failure of Interior Hölder and Gradient Estimates for the Subcritical Special Lagrangian Equation

针对n≥3的次临界特殊拉格朗日方程，研究人员构造显式整体实解析解，证明基于振荡的内部C^α估计失效，且该族解无共同内部连续性模。

凸性与退化特殊 Lagrange 方程

Convexity and the degenerate special Lagrangian equation

本文通过保持时空Lagrange角的坐标变换和部分$C^2$估计，解决DSL最高分支时空凸性及第二分支凸性问题，并揭示其$\star$-乘积结构。

Global Lagrangian solutions of the Camassa-Holm equation

In this paper we construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We first express the equation in Lagrangian flow variable $η$ and then transform it using a change of variable $ρ=\sqrt{η_x}$. The new variable removes the singularity of the CH equation, and we obtain both the global weak conservative solution and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation.

特殊拉格朗日方程的外狄利克雷问题

The exterior Dirichlet problem for special Lagrangian equations

本文针对特殊拉格朗日方程的外狄利克雷问题建立存在唯一性定理，推广了前人工作并改进衰减率，还求解了内狄利克雷问题，将其应用于三维二次Hessian方程的对应问题。

关于广义特殊拉格朗日方程的注记

Notes on generalized special Lagrangian equation

本文通过研究扭曲特殊拉格朗日方程，获得了具有正Lipschitz右端项的Hessian（商）方程的先验$C^{1,1}$估计，并由此证明了3维情形下$σ_2=f^2(x)$粘性解的内部$C^{2,α}$正则性。

Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations

We present a very simple proof of the global existence of a $C^\infty$ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has $C^\infty$ dependence on initial data $u_0$ in the class of $H^s$ divergence-free vector fields for $s>2$.

Global diffeomorphism of the Lagrangian flow-map defining Equatorially trapped water waves

The aim of this paper is to prove that a three dimensional Lagrangian flow which defines equatorially trapped water waves is dynamically possible. This is achieved by applying a mixture of analytical and topological methods to prove that the nonlinear exact solution to the geophysical governing equations, derived by Constantin in J. Geophys. Res., 117 (2012), is a global diffeomorphism from the Lagrangian labelling variables to the fluid domain beneath the free surface.

特殊Lagrangian网络

Special Lagrangian webbing

论文构造了接近正Lagrangian子流形相交点的虚特殊Lagrangian圆柱体家族，获得了开放正Lagrangian子流形的测地线，并首次尝试构建闭正Lagrangian子流形的测地线。同时，提出了一种证明正Lagrangian测地线在非光滑处C^{1,1}正则性的方法。

Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces

We prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with low regularity initial data in Besov spaces $B^{r}_{p,q}(\mathbb{R}^n)$, $r>n/2p$. When $p=2$ and $n\geq 3$, we obtain global solutions, provided the parameters $r,q$ and $n$ satisfy certain inequalities. This is an improvement over known analogous Sobolev space results, which required $n=3$.

Global Solutions to the Lagrangian Averaged Navier-Stokes equation in low regularity Besov spaces

The Lagrangian Averaged Navier-Stokes (LANS) equations are a recently derived approximation to the Navier-Stokes equations. Existence of global solutions for the LANS equation has been proven for initial data in the Sobolev space $H^{3/4,2}(\mathbb{R}^3)$ and in the Besov space $B^{3/2}_{2,q}(\mathbb{R}^3)$. In this paper, we use an interpolation based method to prove the existence of global solutions to the LANS equation with initial data in $B^{3/p}_{p,q}(\mathbb{R}^3)$ for any $p>3$.

Lagrangian approach to global well-posedness of the viscous surface wave equations without surface tension

In this paper, we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab, for which the first complete proof was given in Guo-Tice (2013, Analysis and PDE) via a hybrid of Eulerian and Lagrangian schemes. The fluid dynamics are governed by the gravity-driven incompressible Navier-Stokes equations. Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows, few mathematical works for global existence are based on such an a…

Counterexamples to the comparison principle in the special Lagrangian potential equation

For each $k = 0,\dots,n$ we construct a continuous phase $f_k$, with $f_k(0) = (n-2k)\fracπ{2}$, and viscosity sub- and supersolutions $v_k$, $u_k$, of the elliptic PDE $\sum_{i=1}^n \arctan(λ_i(D^2 w)) = f_k(x)$ such that $v_k-u_k$ has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases $f\colonΩ\to (-nπ/2,nπ/2)$. Our examples show it does not.

A stochastic Lagrangian proof of global existence of the Navier-Stokes equations for flows with small Reynolds number

We consider the incompressible Navier-Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof is based on the stochastic Lagrangian formulation of the Navier-Stokes equations, and works in both the two and three dimensional situation.

Analysis of the One-dimensional Euler-Lagrange equation of continuum mechanics with a Lagrangian of a special form

Flows of one-dimensional continuum in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions. Particular choices of the undefined functions correspond to isentropic flows of an ideal gas, different forms of the hyperbolic shallow water equations. Complete group classification of the equation with respect to these functions is performed. Using Noether's theorem, all conservation laws are obtained. Their analogs in Eulerian coordinates are given.

On hyperbolicity of minimizers for 1D random Lagrangian systems

We prove hyperbolicity of global minimizers for random Lagrangian systems in dimension 1. The proof considerably simplifies a related result in [2]. The conditions for hyperbolicity are almost optimal: they are essentially the same as conditions for uniqueness of a global minimizer in [3].