Solutions of Aronsson equation near isolated points
Vesa Julin
我们证明了对于Aronsson方程的非负解，一个孤立奇点或者是可去除的，或者是解的渐近行为类似于一般锥。推广了Savin，Wang和Yu关于无穷调和函数的渐近性态理论.
We show that for non-negative solution of the Aronsson equation an isolated singularity is either removable, or the solution behaves asymptotically like a general cone. This generalizes the asymptotic behavior theory for infinity harmonic functions by Savin, Wang and Yu.
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Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential
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Isolated singularities of positive solutions of p-Laplacian type equations in R^d
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We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex dimension 2, we show that any complete Kahler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry.
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The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point
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Removable singularity of positive solutions for a critical elliptic system with isolated singularity
研究一类具有临界指数的两耦合椭圆型方程组正奇异解的定性性质。这个系统涉及耦合非线性薛定谔方程的临界指数的非线性光学和玻色-爱因斯坦凝聚。我们证明了解的两个分量的同一个孤立奇点的可去除性的尖锐结果。我们还证明了一个分量在奇异点附近有界而另一个分量在奇异点附近无界的正解的不存在性。这些结果将被应用在随后的工作中，在一个被刺破的球相同的系统将被研究。
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A Parametrization of Solutions to the Yamabe Equation with Isolated Singularity
本文受 Caffarelli 等人工作启发，证明了与调和函数对应的形式无穷级数可产生带不可去孤立奇点的 Yamabe 方程实际解，并提供了构造该类方程在孤立奇点及无穷远处所有解的通用方法。
Asymptotic expansions for conformal scalar curvature equations near isolated singularities
本文研究原点具有孤立奇点的共形标量曲率方程正解的渐近展开，在K满足平坦性条件下建立高阶展开，并给出n≥6时的精细二阶展开，同时将该结果推广到源于Caffarelli-Kohn-Nirenberg不等式的各向异性椭圆方程。
Pointwise Bounds and Blow-up for Choquard-Pekar Inequalities at an Isolated Singularity
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I prove the existence of small time heat expansion for the Laplace operator on an analytic hypersurface with an isolated singularity. First we obtain a local parametrization of the hypersurface near the singularity. We introduce the notion of quasihomogeneous tangent cone. Then perturb the parametrization of the cone employing a Newton scheme and obtain a parametrization with functions of specific form. These allow us to obtain local models for the Laplace operator near the singularity. These are operators with irregular singularities. We derive the estimates required by singular asymptotics.
All $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $(-1)$-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.
On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature
对有界非负曲率函数$-k(z)$在孤立奇点附近，得到了二维曲率方程$Δu=k(z) e^{2 u}$ with $e^{2 u} \in L^1$解的精确渐近性态.
Pointwise Bounds and Blow-up for Systems of Semilinear Elliptic Inequalities at an Isolated Singularity via Nonlinear Potential Estimates
研究了系统$0\leq -Δu\leq f(v)$ $0\leq -Δv\leq g(u)$在$B_1(0)\backslash\{0\}$中的$C^2$正解$u(x)$和$v(x)$在原点附近的行为，其中$f,g:(0,\infty)\to (0,\infty)$是连续函数.我们在$f$和$g$在$\infty$上提供了最优条件，使得该系统的解在原点附近满足逐点界。在维度$n=2$，我们表明，如果$\log^+ f$或$\log^+g$增长最多线性无穷大，此属性成立。在维数$n\geq 3$中，在假设$f(t)=O(t^λ)$，$g(t)=O(t^σ)$为$t\to \infty$，（$λ, σ\geq 0$）的条件下，得到了一条新的临界曲线，它最优地描述了这种逐点界的存在性.我们的方法部分依赖于尖锐的估计自然出现在这种情况下的非线性潜力。
We study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not conformally flat. We prove that, in dimension $6$, any solution is asymptotically close to a Fowler solution, which is an extension of the same result for lower dimensions by F.C. Marques in 2008.
Isolated Boundary Singularities of Semilinear Elliptic Equations
给定一个光滑域$Ω\subset\RR^N$使得$0 \in \partialΩ$和给定$\partialΩ$上的一个非负光滑函数$ζ$，研究了$-Δu=u^q$在$Ω$使得$u = ζ$在$\partialΩ\setminus\{0\}$上的正解在0附近的行为.我们证明了如果$\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$，那么$u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$和我们计算的极限$\abs{x}^{\frac{2}{q-1}} u(x)$ $x \to 0$。我们还调查了案例$q= \frac{N+1}{N-1}$。证明依赖于球域上相关方程解的存在唯一性。
Isolated singularities of 3-dimensional Yang-Mills-Higgs fields
该研究推导了纤维丛上三维Yang-Mills-Higgs场孤立奇点附近的衰减估计，进而得到不同能量条件下的可去奇点定理，推广了经典的三维Yang-Mills场与调和映射的可去奇点定理。
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global …
Minimal surfaces near Hardt-Simon foliations
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Solutions of some Monge-Ampère equations with isolated and line singularities
本文研究了一类具有孤立奇性和线奇性的蒙日-安培方程解的存在性、正则性、分类和渐近性态.我们将$\det \nabla^2 u=1$的所有解分类为具有一个穿刺点的$\R^n$。这可以适用于描述椭球，在相同的精神Serrin的超定问题的拉普拉斯算子。当$k>1$具有$k$不可去奇点时，模仿射等价的所有广义解的集合可以被识别为一个有限维的显式轨道。我们还建立了具有一般奇异集的整体解的存在性、正则性以及广义解在由点或直线组成的奇异点附近的二阶导数的最优估计.几何动机来自奇异半平坦的Calabi-Yau度量。
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