Hessian and gradient estimates for three dimensional special Lagrangian Equations with large phase
Micah Warren, Yu Yuan
本文给出了三维空间中具有至少一个临界值的特殊拉格朗日方程的先验内点Hessian估计和梯度估计。
We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.
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In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions
在一般高维情形下，我们得到了一类特殊的具有临界和超临界相的拉格朗日方程的先验内点Hessian估计。我们的统一的方法导致更尖锐的估计，即使是以前已知的三维和凸解的情况下。
Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow
该研究在超临界拉格朗日相位假设下，得到拉格朗日平均曲率流四类特殊解的内部Hessian估计，并将结果推广至更广泛的拉格朗日平均曲率型方程，丰富了相关几何分析理论。
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。
Hessian estimates for Lagrangian mean curvature equation with Lipschitz critical and supercritical phases
本文提出新策略，结合Allard型正则性定理与几何测度论等方法，在锐利的Lipschitz临界和超临界相位条件下，获得了Lagrangian平均曲率方程解的内部Hessian估计。
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。
Hessian estimates for the sigma-2 equation in dimension three
我们得到了三维特殊拉格朗日方程$σ_{2}=1$的先验内部Hessian估计。
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.
Hessian operators, null Lagrangians and the integral comparison principle
本文研究Hessian算子满足积分比较原理的条件，发现其与零拉格朗日量相关，并完整分类得出仅有k-Hessian线性组合满足该原理，复情形亦类似，揭示多复势理论障碍。
Doubling and the two-dimensional critical valued Lagrangian phase
本文针对二维拉格朗日平均曲率方程，在相变号且梯度沿零集消失时，通过改进的加倍技巧处理退化的Jacobi不等式，建立了内Hessian和梯度估计。
Explicit gradient estimates for minimal Lagrangian surfaces of dimension two
我们推导出明确的，统一的，先验的内部海森和梯度估计的特殊拉格朗日方程的所有阶段在二维。
Hessian estimates for Lagrangian mean curvature equation with sharp Lipschitz phase
本文研究了具有尖锐Lipschitz相位的Lagrangian平均曲率方程，建立了凸解和超临界相位解的先验内部$C^{1,1}$估计，并给出了非Lipschitz情况下的反例，进而推导出$C^0$粘性解的内部$C^{2,\alpha}$正则性。
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。
A special concavity property for positive Hessian quotient operators
该研究建立了正Hessian商算子的特殊凹性性质，并以此证明了黎曼流形上满足对应方程的一般对称张量的Jacobi不等式，丰富了相关算子的性质研究。
An ansatz for constructing explicit solutions of Hessian equations
本研究提出一种二次曲面变体拟设，可将多类复/实Hessian方程约化为可积常微分方程组，构造出任意次临界相位的全局显式解，并验证特殊拉格朗日爆破解与已知结果一致。
In this paper, we prove some rigidity theorems for the entire 2-convex solutions of 2-Hessian equation in Euclidean space. As an application, we obtain a Bernstein type theorem for global special Lagrangian graphs.
In this paper, we prove interior gradient estimates for the Lagrangian mean curvature equation, if the Lagrangian phase is critical and supercritical and $C^{2}$. Combined with the a priori interior Hessian estimates proved in [Bha21, Bha22], this solves the Dirichlet boundary value problem for the critical and supercritical Lagrangian mean curvature equation with $C^0$ boundary data. We also provide a uniform gradient estimate for lower regularity phases that satisfy certain additional hypotheses.
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.
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。
Variational integrals on Hessian spaces: partial regularity for critical points
本研究建立Hessian空间中变分积分临界点的正则性理论，证明满足小BMO条件的有界Hessian临界点的光滑性，给出奇异集Hausdorff维数上界，并应用于拉格朗日几何变分问题。
A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.
In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $\mathbb{R}^n$.
Global solutions to special Lagrangian equations
我们证明了相位大于临界值的特殊拉格朗日方程的任何整体解必定是二次的。
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。
Special Lagrangian submanifolds in the complex sphere
构造了复球面上一族具有SO（n）-不变性的拉格朗日子流形.其中，我们发现那些是特殊的拉格朗日方面的Calabi-Yau结构所定义的Stenzel度量。
Singular Solutions to Special Lagrangian Equations with Subcritical Phases and Minimal Surface Systems
我们构造了具有次临界相的特殊拉格朗日方程和极小曲面系统的奇异解。相应地给出了光滑解的先验估计破缺族.具有一定凸性的特殊拉格朗日方程的先验估计已得到广泛的研究。
A Liouville type theorem for special Lagrangian Equations with constraints
对于具有一定“凸性”的特殊拉格朗日方程，在解的约束线性增长假设下，得到了一个Liouville型结果.
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.
A Bernstein problem for special Lagrangian equations
对这类特殊的拉格朗日方程，我们得到了一个Bernstein型结果，即任何整体凸解必是二次的。从极小曲面的角度，证明了任何具有凸势的全局极小拉格朗日图必是一个超平面。
In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.
Singular Solution to Special Lagrangian Equations
在非凸情形下，证明了一类特殊拉格朗日方程非光滑解的存在性。
A Bernstein type result for special Lagrangian submanifolds
设R是\C^n的完备极小拉格朗日子流形.我们确定区域的格拉斯曼的拉格朗日子空间，使每当图像的高斯映射的图像的拉格朗日位于这些地区之一，然后是一个仿射空间。
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。
Minimal surface systems, maximal surface systems, and special Lagrangian equations
本文推广了Osserman引理关于高余维二维极小图的广义Gauss映射，明确构造了一个Jenkins-Serrin型的特殊Lagrangian Scherk图，推广了Calabi关于极小图与极大图的对应关系.
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.
Deformations of Asymptotically Conical Special Lagrangian Submanifolds
麦克莱恩研究了变形紧凑的特殊拉格朗日子流形，特别表明，他们来在模空间的尺寸只取决于拓扑的子流形。在这篇文章中，我们研究了类似的问题，非紧的，“渐近圆锥”SL子流形，关于各种“边界条件在无穷远”。
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
设$M_{1}$和$M_{2}$是$\Rtn$的两个特殊的拉格朗日子流形，其边界横交于一点$p$ .集合$M_{1} \cup M_{2}$是一个奇异的特殊拉格朗日簇，在交点有一个孤立的奇点。进一步假设相交处的切平面满足角度条件（在维度$n=3$中始终成立）。那么，$M_{1} \cup M_{2}$是可正则的;换句话说，存在一族光滑的极小拉格朗日子流形$M_α$，其边界在适当的拓扑中收敛到$M_{1} \cup M_{2}$。这个结果是通过首先将光滑颈粘到$M_{1} \cap M_{2}$的邻域中，然后通过扰动这个近似解直到它变得最小和拉格朗日而获得的。
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $α'$. Actually we obtain at the same time a solution of the open case $α'>0$, an improved solution of the known case $α'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $Γ_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.
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。
In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented Hessian is only assumed regular. This enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case. Finally we give some applications to existence theorems which embrace st…
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.
Lagrangian aspects of the axisymmetric Euler equation
本文研究了三维轴对称圆柱体上有旋欧拉方程组解的几何性质。写的方程在拉格朗日形式的流动导数沿轴或边界和施加奇数的垂直分量的流量，我们扩展了一些爆破标准，由于蔡，康斯坦丁和吴有关的假设的符号上的压力海森。此外，我们给出了一个几何解释的结果，无论是在沿轨迹的局部几何和黎曼几何的体积保持双态群。
Special Lagrangian cones with higher genus links
对每一个奇自然数g=2d+1，利用几何PDE胶合方法，证明了亏格为g的闭黎曼曲面上C^3中存在可数无限族特殊拉格朗日锥.
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。
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。
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…
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.
Liouville theorem for a class of Hessian equations
本文研究包含 Monge-Ampère 方程等在内的广义 Hessian 椭圆方程，提出解的新附加条件并证明 Liouville 定理，将文献中多个孤立的 Liouville 定理和条件内部估计进行了重要推广。
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.
We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form $$f(λ(D^2u))=g(x),$$ with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli--Nirenberg--Spruck \cite{Caffarelli1985}, Trudinger \cite{Trudinger1995} and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that $f$ is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniquen…
Dirichlet problem for degenerate Hessian quotient type curvature equations
本文研究退化Hessian商型曲率方程的Dirichlet问题，在特定正则性条件下证明 $C^{1,1}$ 图形超曲面的存在唯一性，并给出退化Hessian型方程在最优条件下的二阶导数估计。
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.
We study the long-time behavior of solutions of the $k$-Hessian evolution equation $u_t=S_{k}(D^2 u)$, posed on a bounded domain of the $n$-dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of $u$ is precisely described by this special solution. Further, we initiate the study of that dynamic phenomenon on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call $k$-Barenblatt solutions. These solutions present some common properties as those of well…
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