Zhisu Li
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.
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We consider the exterior Dirichlet problem for Monge-Ampère equation with prescribed asymptotic behavior. Based on earlier work by Caffarelli and the first named author, we complete the characterization of the existence and nonexistence of solutions in terms of their asymptotic behaviors.
On the exterior Dirichlet problem for Hessian equations
本文建立了在无穷远处具有规定渐近行为的Hessian方程的外狄利克雷问题解的存在性定理。这将Caffarelli和Li关于蒙日-安培方程的一个结果推广到Hessian方程。
In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(λ(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior at infinity.
In this paper, we study the Dirichlet problem of Hessian quotient equations in exterior domains. By estimating the eigenvalues of the solution, the necessary and sufficient conditions on existence of radial solutions are obtained. Applying the solutions of ODE, the viscosity subsolutions and supersolutions are constructed and then the existence of viscosity solutions for exterior problems is established by the Perron's method.
Exterior Dirichlet problem for Hessian equations on a non-convex ring
本文证明了非凸环上Hessian方程外Dirichlet问题光滑解的存在性，拓展了Bao-Li-Li在该问题上的已有研究结果。
Explicit Formula of the Infinite Energy Solutions for the Exterior 2D Div-Curl Problem with Dirichlet Boundary Condition
本文推导了外部二维div-curl问题在狄利克雷边界条件下无穷能量解的显式公式。
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.
In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-β})$ near infinity, for some $β>2$. Under a prescribed generalized symmetric asymptotic behavior at infinity, we establish an existence and uniqueness theorem for viscosity solutions, by using comparison principles and Perron's method. This extends the previous results for Monge--Ampère equations and Hessian equations.
A mixed variational formulation of some problems in $L^2$-based Sobolev spaces is used to define the Newtonian and layer potentials for the Stokes system with $L^{\infty}$ coefficients on Lipschitz domains in ${\mathbb R}^3$. Then the solution of the exterior Dirichlet problem for the Stokes system with $L^{\infty}$ coefficients is presented in terms of these potentials and the inverse of the corresponding single layer operator.
In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain $\mathbb{C}^n\setminusΩ$. We prove the existence and uniqueness of the $C^{1,1}$ solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second order estimate.
This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem. Applications of these estimates to bilinear type commutator estimates for Laplace operator with Dirichlet boundary condition in exterior domain are discussed too.
Exterior Dirichlet Problems for Hessian Quotient Equations of Mixed Type
本研究针对混合型Hessian商方程的外狄利克雷问题，通过三阶段次解构造证明了具有指定二次无穷远渐近行为的光滑容许解的存在与唯一性，并得到余项的全导数衰减。
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…
Monge-Ampère equation $\det(D^2u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Ampère equation defined either globally in $\mathbb R^2$ or outside a convex set. First we prove the existence of a global solution that satisfies a prescribed asymptotic behavior at infinity, if $f$ is asymptotically close to a positive constant. Then we solve the exterior Dirichlet problem if data are given on the boundary of a convex set and at infinity.
In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze …
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。
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.
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $Ω$, $u=g$ in $\mathbb R^N\setminusΩ$, in non-smooth domains $Ω$. When $g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right hand side, for which the boundary regularity is well understood. Here, we study the case in which $g\in C^{0,α}$, and establish the optimal Hölder regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $Ω$.
Existence and Nonexistence for Hessian Exterior Dirichlet Problems with \(k\)-Admissible Asymptotic Matrices
研究k-Hessian方程的外部Dirichlet问题，允许渐近矩阵仅为k-容许而不必正定，通过线性化得到正确度量，构造外部屏障和子解，并利用线性化容量比较和切向迹矛盾完成存在性与不存在性的刻画。
Existence of classical solutions to the exterior Dirichlet problem for Hessian quotient equations
本文针对右端非恒定的Hessian商方程外狄利克雷问题，通过构造半径无关的边界到内部估计与爆破论证，证明了经典容许解的存在性及无穷远处Hessian的收敛性，还处理了源项非径向扰动等情形。
In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations: \begin{equation}\label{GPDE0} (-Δ)^{\fracα{2}}u(x)=f(x,u) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\,\, Ω_{r}:=\{x\in\mathbb{R}^{n}\,|\,|x|>r\} \end{equation} with arbitrary $r>0$, where $n\geq2$, $0<α\leq 2$ and $f(x,u)$ satisfies some assumptions. A typical case is the Hardy-Hénon type equations in exterior domains. We first derive the equivalence between \eqref{GPDE0} and the corresponding integral equations \begin{equation}\label{GIE0} u(x)=\int_{Ω_{r}}G_α(x,y)f(y,u(y)…
Two Necessary and Sufficient Conditions to the Solvability of the Exterior Dirichlet Problem for the Monge-Ampère Equation
本文研究Monge–Ampère方程外部Dirichlet问题，借助光滑逼近与Perron方法给出可解性的两个充要条件，并放宽了边值和内边界的正则性假设。
Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data
我们考虑三维空间中单位球外部的散焦能量临界非线性系统。在索伯列夫空间$\dot H_0^1$中，对Dirichlet边界条件下的初值问题，证明了该方法的全局适定性和大径向初值的离散性.我们还指出，同样的策略可以用来处理球外部的Dirichlet边界条件和径向$\dot H_0^1$初始数据的能量-超临界NLS。
Large-time estimates for the Dirichlet heat equation in exterior domains
本文研究了外区域中 Dirichlet 热方程的大时间渐近行为，给出了一致范数和 $L^1$ 范数下的显式逼近速率估计，并利用熵方法揭示了热核的渐近轮廓及其与区域形状和质量损失的关系。
The exterior Dirichlet problem for the homogeneous $k$-Hessian equation
本文研究齐次 $k$-Hessian 方程的外部 Dirichlet 问题，通过构造逼近解并利用一致 $C^{1,1}$ 估计证明了存在性、唯一性及正则性，并导出了加权几何不等式。
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。
The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore--Gibson--Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition.
In this paper we study the dependence of the Hölder estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the Hölder regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), from which we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).
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.
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。
On the existence of solutions to the planar exterior Navier Stokes system
本文考虑定常不可压Navier斯托克斯方程在圆盘B外的解，圆盘上的Dirichlet边界条件为非零，无穷远处的边界条件为零.证明了无对称边界条件的开集解的存在性。
We study the boundary value problem for the stationary Navier--Stokes system in two dimensional exterior domain. We prove that any solution of this problem with finite Dirichlet integral is uniformly bounded. Also we prove the existence theorem under zero total flux assumption.
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.
Perron solutions and boundary regularity for nonlocal nonlinear Dirichlet problems
研究分数\(p -\)拉普拉斯型非线性算子非局部狄利克雷问题，考虑索伯列夫解与佩龙解，证明二者对应的边界正则性概念等价，给出佩龙解新定义及相关可解性、唯一性结果，揭示其在特定扰动下的不变性。
Capacitary estimates for solutions to nonlocal Dirichlet problems
该研究针对带界可测系数的非局部非线性椭圆方程弱解，建立了容量密度条件与一致边界赫尔德估计的等价性，并推导了刻画正则性传递的精细容量估计。
We prove global well-posedness of the two-dimensional exterior Navier-Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions for asymptotically constant initial data and arbitrary large Reynolds numbers.
We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.
A Serrin-type over-determined problem for Hessian equations in the exterior domain
研究外区域Hessian方程的超定问题，证明存在唯一有界区域使问题有唯一严格凸解。
Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols
该论文研究由非对称 Lévy 算子驱动的半线性外部 Dirichlet 问题，在符号逐点收敛条件下证明解的稳定性，涵盖局部、非局部及混合算子，并给出对称情形下部分形式的 Mosco 收敛结果。
We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in $\mathbb R^N$, and equipped with homogeneous Dirichlet boundary conditions.
$L^2$ decay estimates of weak solutions to 3D fractional MHD equations in exterior domains
本文研究了在Dirichlet边界条件下外部区域中三维分数阶MHD方程弱解的渐近行为，并获得了弱解的$L^2$衰减估计。
The Dirichlet Problem for Lévy-stable operators with $L^2$-data
本文证明了 $2s$-稳定过程生成元在加权 $L^2$ 空间粗糙外部数据下 Dirichlet 问题分布解的 Sobolev 正则性，该算子类包含分数阶 Laplacian，且其正则性估计在 $s\ o 1-$ 极限下稳健以恢复局部理论。
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.
Regular solutions to a supercritical elliptic problem in exterior domains
考虑Dirichlet条件为零的外区域$Ω= \mathbb{R}^N \setminus D$中的超临界椭圆问题-Δu = λe^u，λ> 0，其中D在\mathbb{R}^N中光滑有界，N大于或等于3.我们证明了，当λ很小时，这个问题有无穷多个正则解.
Ill-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equation
本文研究了二维拉格朗日均曲率方程狄利克雷问题的不相容性，通过纳什-库伊珀构造得到无限多个非常弱解，并指出这些解无法提升为更高正则性解。
It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvabil…
A Pöschl--Teller approximation to the wave equation outside a torus
针对三维环面外部波动方程时谐问题无法分离变量的难点，构建与Pöschl-Teller势薛定谔算子相关的近似算子，结合Mehler-Fock核与已知本征解完成高频分析，得到环面附近的紧密色散估计。
The sharp lifespan of small data smooth solutions to 2-D quadratic quasilinear wave equations in exterior domains
针对二维外区域中具有齐次Dirichlet或Neumann边界条件的二次拟线性波动方程，证明了光滑解的生命跨度下界为C/ε²，并给出了径向对称解的上界及尖锐常数。
The inhomogeneous fractional Dirichlet problem
该研究针对广义Hölder空间中2s-稳定算子的非齐次Dirichlet问题，采用直接求解而非外部数据延拓相减的方法，得到边界正则性结果，并用反例证明其最优性，该结果对分数阶Laplacian也属新成果。
Self-similar asymptotics of solutions to the Navier-Stokes system in two dimensional exterior domain
本文研究了二维不可压Navier-斯托克斯方程在一个障碍物外的Dirichlet边界条件。假设速度在无穷远处的环量足够小，我们证明了相应的初边值问题解的大时间行为是由Lamb-Oseen涡描述的.后者是著名的Navier-Stokes方程组在全空间的显式自相似解$\R^2$。
Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain
本文针对二维外部区域中临界半线性阻尼波动方程，在径向对称条件下，证明了其生命跨度与临界半线性热方程相同的双指数下界和上界估计，并引入了控制L^1范数的新技术及带对数权重的Gagliardo-Nirenberg型估计。
Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain
考虑三维区域上半线性波动方程的Cauchy-狄利克雷问题，该区域位于有界非捕获障碍物的外部。我们得到了详细的估计寿命的下界的经典解时，初始数据的大小趋于零，在一个类似的精神，约翰和Hörmander的作品柯西问题的治疗。我们证明了我们的估计至少在某些特殊情况下是尖锐的。
Non-autonomous Ornstein-Uhlenbeck equations in exterior domains
本文考虑光滑外区域$Ω\subset \R^d$上的非自治Ornstein-Uhlenbeck算子在Dirichlet边界条件下的解.在适当的系数假设下，相应的非自治抛物柯西问题的解由一个关于$1< p < \infty$的$L^p(Ω)$的演化系统$\{P_Ω(t,s)\}_{0\leq s\leq t}$控制.进一步得到了空间导数的$L^p$ -估计和$P_Ω(t,s)$的$L^p$ - $L^q$光滑性质，即$0\leq s\leq t$.
Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains
本文研究了一类完全非线性椭圆算子的狄利克雷问题的最大值原理、特征值的存在性和解的存在性，其中算子具有与p-Laplacian算子相似的奇异性或退化性，其新颖之处在于我们考虑了有界区域上的方程，它们只满足外锥条件.
Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains
本文证明了一类具有二次非线性项的多速Dirichlet波动方程在障碍物外解的整体存在性。我们假设多速度拟线性波动方程组的自然零条件。零条件只限制了每个波族的自相互作用。我们使用交换向量场和加权时空$L^2$估计的方法。
In this paper we investigate the uniqueness of solutions of the steady planar Navier-Stokes equations with different boundary conditions in the exterior domain. For a class of incompressible flow with constant vorticity, we prove the uniqueness of the solution under the enhanced Navier boundary conditions. At the same time, some counterexamples are given to show that the uniqueness of the solution fails under the Navier boundary conditions. For the general incompressible flow with Dirichlet boundary condition, we prove various sufficient conditions for the uniqueness of the solution.
Asymptotics of solutions to the Navier-Stokes system in exterior domains
本文研究了在具有$n\geq2$的$\mathbb{R}^n$的外部区域上具有Dirichlet边界条件的不可压Navier-Stokes方程.我们比较长时间的行为的解决方案，这个初边值问题的长期行为的解决方案的类似柯西问题在整个空间$\mathbb{R}^n$。我们发现，无论是在弱$L^{n}$-空间中的小初始数据的情况下，还是对于某一类大初始数据，这两个问题的解的长时间渐近性都是一致的。
In this paper, we study the exterior problem for the maximal surface equation. We obtain the precise asymptotic behavior of the exterior solution at infinity. And we prove that the exterior Dirichlet problem is uniquely solvable given admissible boundary data and prescribed asymptotic behavior at infinity.
The Behavior of the Free Boundary for Reaction-Diffusion Equations with Convection in an Exterior Domain with Neumann or Dirichlet Boundary Condition
设\begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} \end{equation*}是二阶椭圆算子，考虑具有Neumann边界条件的反应扩散方程，\begin{equation*} \begin{aligned} &Lu=Λu^p\ \text{in}\ \mathbb{R}^d-D;\\ &\nabla u\cdot \bar n=-h\ \text{on}\ \partial D;\\ &u\ge0 \ \text{is minimal}, \end{aligned} \end{equation*}，其中$p\in(0,1)$，$d\ge2$，$h$和$Λ$是连续正函数，$D\subset R^d$是有界的，$\bar n$是区域$\mathbb{R}^d-\bar D$的向内法线.还考虑用狄利克雷边界条件代替诺依曼边界条件的相同方程，即$\partial D$上的$u=h$。上述方程的解可能具有一个自由边界。当$D=\{|x|<R\}$和$L$和$Λ$径向对称时，我们将解写成$u(r)$和$r=|x|$，并通过$r^*(h)=\inf\{r>R:u(r)=0\}$定义自由…
We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and an exterior determination method from the partial exterior…
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