Zixiao Liu, Jiguang Bao
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.
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We studied the asymptotic behavior of solutions with quadratic growth condition of a class of Lagrangian mean curvature equations $F_τ(λ(D^2u))=f(x)$ in exterior domain, where $f$ satisfies a given asymptotic behavior at infinity. When f(x) is a constant near infinity, it is not necessary to demand the quadratic growth condition anymore. These results are a kind of exterior Liouville theorem, and can also be regarded as an extension of theorems of Pogorelov, Flanders and Yuan.
Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow in $\mathbb R^{n+1}_-$
本文研究Lagrangian平均曲率流古代解在无穷远处的渐近行为，证明其收敛于二次多项式与线性函数之和，并建立适用于所有维数且无需正定条件的全局粘性解存在性，揭示了刚性、渐近与粘性解之间的联系。
Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains
研究外域上超临界拉格朗日均曲率方程解在无穷远处的渐进行为，采用尺度依赖的差商方法和非局部势方法，无需对f二次求导，仅在Lipschitz正则性下获得定量Hessian收敛，得到所有维度和所有衰减率β>0的最优渐进行为，包括临界对数情况。
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.
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。
In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.
We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Ampère type equations that origin from mean curvature equations of Lagrangian graph $(x,Du(x))$ and refine the previous study on zero mean curvature equations and the Monge-Ampère equations.
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。
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.
Lagrangian Mean Curvature Equations on exterior domains
本文提出扩展的外域(K,K′,α₀)拟共形映射方法，研究超临界相拉格朗日均曲率方程在无穷远处的渐进行为，扩展了Li-Li-Yuan关于特殊拉格朗日方程的外域伯恩斯坦型定理。
We continue our study of tempered oscillatory integrals $I_φ(a)$, here investigating the link with a suitable symplectic structure at infinity, which we describe in detail. We prove adapted versions of the classical theorems, which show that tempered distributions of the type $I_φ(a)$ are indeed linked to suitable Lagrangians extending to infinity, that is, extending up to the boundary and in particular the corners of a compactification of $T^*\mathbb{R}^d$ to $\mathbb{B}^d\times\mathbb{B}^d$. In particular, we show that such Lagrangians can always be parametrized by non-homogeneous, regular p…
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.
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。
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。
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。
We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.
Special Lagrangian pair of pants
本文在 $T^n$ 的余切丛中构造了一般维数的特殊拉格朗日裤子。
We construct a family of Lagrangian submanifolds in the complex sphere with a SO(n)-invariance property. Among them we find those which are special Lagrangian with respect with the Calabi-Yau structure defined by the Stenzel metric.
We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.
We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.
We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.
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.
We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.
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。
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.
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".
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。
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.
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.
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。
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.
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.
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。
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.
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。
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.
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.
Consider any Leray-Hopf weak solution of the three-dimensional Navier-Stokes equations for incompressible, viscous fluid flows. We prove that any Lagrangian trajectory associated with such a velocity field has an asymptotic expansion, as time tends to infinity, which describes its long-time behavior very precisely.
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…
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。
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。
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。
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。
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。
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。
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…
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。
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 …
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.
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