Mu-Tao Wang
We solve the Cauchy-Dirichlet problem for the minimal surface system in arbitrary dimension and codimension assuming a condition on the variation of the initial submanifold .
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We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian type ambient spaces, and obtain general existence and uniqueness results which apply to any codimension.
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.
Liouville theorem for immersed minimal surfaces in any codimension
研究了浸入最小盘在余维数任意情况下的李维定理，证明了谐函数的负部分以亚线性速率增长时为常数，从而得出更高余维数的伯恩斯坦定理，并通过猫眼曲面、螺旋面和恩内珀族曲面证明结果最优，同时展示了谐函数的均匀霍尔德正则性。
On the non-existence of solutions of the Dirichlet problem for the minimal surface system
本文研究了最小曲面狄利克雷问题解的不存在性，包括在非C¹边界下的情况。
Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.
We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.
In this paper, we prove that the infimum of the mean curvature is zero for a translating solitons of hypersurface in $\re^{n+k}$. We give some conditions under which a complete hypersurface translating soliton is stable. We show that if the norm of its mean curvature is less than one, then the weighted volume may have exponent growth. We also study the Dirichlet problem for graphic translating solitons in higher codimensions.
We study the problem of finding a minimal graph with prescribed boundary data in arbitrary dimension and codimension. Existence, uniqueness, stability and regularity are treated. We first present the well-known results for codimension one: Jenkins-Serrin's existence theorem, convexity properties of the area which give uniqueness and stability and De Giorgi's theorem for regularity. In higher codimension we first discuss the counterexamples of Lawson and Osserman. Then we present the recent results of Mu-Tao Wang: we use the mean curvature flow to show existence for small boundary data and we p…
We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson-Osserman philosophy.
Calibrating Forms for Minimal Graphs in Arbitrary Codimension
本文引入了一种与欧几里得空间中最小图形子流形自然相关的闭合微分形式，通过构造特定形式来校准最小图，从而推导出新的校准最小图，并验证了Lawson和Osserman的猜想。
In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $Σ$ is the graph of a (strictly) distance-decreasing map, then $Σ$ is (strictly) stable. It is known that a minimal graph of codimension one is stable without assuming the distance-decreasing condition. We give another criterion for the stability in terms of the two-Jacobians of the map which in particular covers the codimension one case. All…
We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dimensional planes, and in a neighbourhood of any point at which at which at least one tangent cone is equal to a union of four distinct multiplicity one n-dimensional half-planes that meet along an (n-1) - dimensional axis. The key ingredient is a new Excess Improvement Lemma obtai…
The maximal graph Dirichlet problem asks whether there exists a spacelike graph, in a semi-Euclidean space, with a given boundary and with mean curvature everywhere zero. We prove the existence of solutions to this problem under certain assumptions on the given boundary. Most importantly, the results proved here will hold for graphs of codimension greater than 1.
We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation of the author. The paper is organised as follows. First we introduce Cartan-Hadamard manifolds and the concept of asymptotic Dirichlet problem, then discuss about the development of the results and describe the methods used in the proofs. In the end we mention some results about the nonsolvability of the asymptotic Dirichlet problem.
Existence and non-existence of minimal graphs
本文通过平均曲率流研究任意维数余维数下极小曲面系统的Dirichlet问题，证明了平均凸有界域上极小图的存在性，推广了Jenkins-Serrin判据，并构造了余维数2不可解的反例，还探讨了唯一性。
In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane.
Stability of Diophantine Minimal Foliations in Arbitrary Codimension
该研究将Moser的极小超曲面叶状结构稳定性定理推广到任意余维，证明标准环面上满足丢番图条件的平面诱导的仿射叶状结构，在欧氏度量的小周期扰动下仍以极小子流形构成的光滑共轭叶状结构存在，采用Hamilton的Nash-Moser隐函数定理完成证明。
Let Σbe a minimal submanifold of \R^{n+m} that can be represented as the graph of a smooth map f:\R^n-->\R^m. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which Σmust be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.
Topology and Dirichlet spectrum of free boundary minimal and CMC surfaces
本文在凸边界且非负Ricci曲率的三维流形中，对指标为一的自由边界极小曲面或稳定的自由边界常平均曲率曲面，在Jacobi算子第一Dirichlet特征值非负时，给出亏格和边界分量数的界，并应用于三维正规齐次空间的严格凸区域，得到强拓扑限制。
The Dirichlet problem for the minimal surface system on smooth domains
本文提出涉及小振荡与 $C^2$ 范数的新假设，在非负 Ricci 曲率区域上通过带边界的平均曲率流建立极小曲面系统的 Dirichlet 问题，利用高余维自收缩子的 Bernstein 型定理证明流的长时间存在性，并因不限制区域直径推出外 Dirichlet 问题的存在性结果。
Constructing entire minimal graphs by evolving planes
本文提出通过演化平面方法构建高维极小图，并揭示其与特殊拉格朗日图的关系。
In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second author shows that two-valued $C^{1, μ}$ harmonic functions on $n$ dimensional domains, which are typically-non-minimizing stationary points of Dirichlet energy, play an essential role in the study of multiplicity 2 branch points of stable codimension 1 rectifiable currents of dim…
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.
Solutions to the minimal surface system with large singular sets
本文构造了最小曲面系统 Dirichlet 问题中 Lipschitz 图极小化序列的 Cartesian current 极限，证明其可具有大的内部垂直和非极小部分，揭示了高余维下参数化与非参数化面积极小化问题的显著差异，且构造达到最小维数 n=3 和余维数 m=2。
Let $L$ be a special Lagrangian submanifold of a compact, Calabi-Yau manifold $M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is shown how deformations of $L$ which keep the boundary of $L$ confined to $W$ can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near $L$ with boundary on $W$ is found to be finite dimensional and is parametrised over the space of harmonic 1-forms of $L$ satisfying Neumann boun…
Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.
The Dirichlet problem for the minimal surface equation on unbounded helicoidal domains of $\mathbb{R}^{m}$
该研究针对$\u211d^{n+1}$中螺旋投影为外域的无界$G$-不变$C^{2,\u03b1}$-域，证明了零边界极小曲面Dirichlet问题存在一族梯度边界上确界为$|s|$的$G$-不变解，并补充了解的无穷远行为信息。
$L^p$-Sobolev inequalities on minimal submanifolds
本文通过最优质量传输理论，证明了在欧几里得最小子流形中任意余维数情况下具有显式常数的$L^p$-Sobolev不等式，并提供了Brendle和Eichmair周长不等式的新证明方法。
An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow
本文提出了一种适用于非均凸域的极小曲面方程可解性准则，并将其应用于均曲率流，解决了传统方法无法处理的某些情况。
In this paper, we study large $m$ asymptotics of the $l^1$ minimal $m$-partition problem for Dirichlet eigenvalue. For any smooth domain $Ω\in \mathbb{R}^n$ such that $|Ω|=1$, we prove that the limit $\lim\limits_{m\rightarrow\infty}l_m^1(Ω)=c_0$ exists, and the constant $c_0$ is independent of the shape of $Ω$. Here $l_m^1(Ω)$ denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any $m$-partition of $Ω$.
In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing $n$-dimensional rectifiable current of arbitrary codimension. In either case, the dimension bound is sharp. We develop estimates to study the asymptotic behaviour of a multi-valued Dirichle…
In this paper we study the Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds with dimension $n$. Imitating the generalized solution theory of Miranda-Giusti, we define a new conformal area functional and a generalized solution to this Dirichlet problem. The existence of generalized solutions to this problem on bounded Lipschitz domains is established. If the domain is mean convex and bounded with $C^2$ boundary, its closure does not contain any closed minimal hypersurface except a singular set with its Hausdorff dimension at most $n-7$ and the bound…
We consider a Plateau problem in codimension $1$ in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary $\partial Ω$ of a bounded convex domain $Ω\subset\mathbb{R}^2$. Where the Dirichlet datum is not prescribed, we allow a free contact with the horizontal plane. We show existence of a solution, and prove regularity for the corresponding minimal surface. Finally we compare these solutions with the classical minimal surfaces of Meeks and Yau, and show that they are equivalent when the Dirichlet boundary datum is assigned in at most $2$ disjoint arcs of $…
For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.
Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds
本文在紧致流形上构造了磁性 Ginzburg-Landau 方程的解，使其集中于定向非退化的余维2极小子流形附近，在远离子流形处具有精确渐近形态，改进了此前仅在测度意义成立的集中现象结果。
Practical tensor calculus on embedded submanifolds of arbitrary codimension
提出一种完全外蕴、无参数化的张量微积分框架，适用于任意维数和余维的嵌入子流形，涵盖切向投影、外蕴与协变导数、外蕴Stokes公式及Laplace-Beltrami算子，并通过三个独立应用展示其计算与理论分析能力。
Anisotropic minimal surface equation with Dirichlet boundary condition
研究有界区域中各向异性极小曲面方程狄利克雷问题，通过建立先验梯度估计等方法，证明其唯一可解性、广义极小值的相关性质，还解释了变分形式产生诺伊曼边界条件及该问题无需边界曲率约束的原因。
We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain Adams' criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Most of the results are new for open E (apart from those which are trivial in this case) and also on R^n.
It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg-Landau equations $-Δu_\varepsilon +\varepsilon^{-2}(|u_\varepsilon|^2-1)u_\varepsilon = 0$, the energy and vorticity concentrate as $\varepsilon\to 0$ around a codimension $2$ stationary varifold -- a (measure theoretic) minimal surface. Much less is known about the question of whether, given a codimension $2$ minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for mi…
We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.
Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces
研究紧致加倍度量测度空间中反射分数阶 p-Laplace 型非齐次方程的 Dirichlet 问题，证明了存在性、唯一性、比较与稳定性，并建立了内部正则性和 Kellogg 型性质。
We consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane. We prove the existence of a minimal solution to this equation and study its properties. We infer from this analysis some asymptotics for the stationary solution of an equation arising in plasma physics.
On generalized Dirichlet integrals in the smooth and in the o-minimal setting
本文研究紧流形上广义狄利克雷能量对应的集合在L²(M)中的预紧性问题，基于迭代特征集给出几何充分条件，证明驯服向量场在稠密点满足某阶Hörmander条件时，预紧性仅受特征子流形阻碍，并探讨了平方和算子全局正则性的推论。
This paper studies the Dirichlet problem for Laplace's equation in a domain $Ω_{\varepsilon, η}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $η$ the ratio between the scale of sizes of holes and $\varepsilon$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on the small parameters $\varepsilon$ and $η$. We also show that these estimates are either optimal or near optimal.
We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction having the combined effects of a singular term and of a parametric $(p-1)$-superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies. Moreover, we prove the existence of a minimal positive solution $u^*_λ$ and study the monotonicity and continuity properties of the map $λ\to u^*_λ$.
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