Liliane A. Maia, Benedetta Pellacci
The existence of a positive solution for a class of asymptotically lin- ear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function.
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In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric…
We verify the three-dimensional Glassey conjecture for exterior domain (M, g), where the metric g is asymptotically Euclidean, provided that certain local energy assumption is satisfied. The radial Glassey conjecture exterior to a ball is also verified for dimension three or higher. The local energy assumption is satisfied for many important cases, including exterior domain with nontrapping obstacles and flat metric, exterior domain with star-shaped obstacles and small asymptotically Euclidean metric, as well as the nontrapping asymptotically Euclidean manifolds (R^n, g).
Nonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains
本文研究ℝⁿ（n≥1）外区域中非线性分数阶椭圆不等式正解的不存在性，在极弱非线性项假设下得到新结果，且该结论在全空间情形下也具新颖性。
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive s…
We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass $$-Δu = f(u), \qquad u\in D_0^{1,2}(Ω_R),$$ where $Ω_R:=\{x \in \mathbb{R}^N:|u|>R\}$ with $R>0$, $N\geq4$, and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. We show that the number of positive multipeak solutions becomes arbitrarily large as $R \to \infty$.
We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying boundary data, for $t \in R$ or $t \in (0,\infty)$. In the latter case it is coupled with small initial data in weak $L^{3}$. As a corollary, the unique existence of time-periodic solutions is shown for the small periodic boundary data. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically discrete…
We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-Δ_p u-\fracμ{|x|^p}u^{p-1}=\frac{C}{|x|^σ}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $μ\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,σ)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator wi…
Strong solutions of fractional Boussinesq equations in an exterior domain
本文研究三维球外区域中由加热引起的热对流问题，通过谱表示建立分数阶Boussinesq方程系统，证明Hilbert空间中小强解的存在性，并由此得到稳态的局部稳定性与渐近吸引性。
We consider the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. We assume that the initial velocity is a finite-energy and L^q-summable perturbation of the Oseen vortex with circulation α, where 1 < q < 2. If α is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation α. This is a global stability result, which holds for arbitrarily large perturbations of the Oseen vortex, and our smallness assumption on the circulation is independent of the domain under cons…
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.
We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as $|x|$ tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded $p$-harmonic function is constant if $1<p<d$. If $p\ge d$, then it is either constant or it behaves asymptotically like the fundamental solution of the homogeneous $p$-Laplace eq…
{\bf Abstract} \,\, We consider the following nonlinear Schrödinger equation on exterior domain. \begin{equation} \begin{cases} iu_t+Δ_g u + ia(x)u - |u|^{p-1}u = 0 \qquad (x,t) \in Ω\times (0,+\infty), \qquad (1)\cr u\big|_Γ= 0\qquad t \in (0,+\infty), \cr u(x,0) = u_0(x)\qquad x \in Ω, \end{cases} \end{equation} where $1<p<\frac{n+2}{n-2}$, $Ω\subset\mathbb{R}^n$ ($n\ge3$) is an exterior domain and $(\mathbb{R}^n,g)$ is a complete Riemannian manifold. We establish Morawetz estimates for the system (1) without dissipation ($a(x)\equiv 0$ in (1)) and meanwhile prove exponential stability of th…
Existence of a positive solution for a class of Schrödinger logarithmic equations on exterior domains
本文针对外部区域上一类Schrödinger对数方程，提出新方法使C¹变分方法适用，证明了正解的存在性。
The paper concerns with positive solutions of problems of the type $-Δu+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $Ω\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $Ω$ can be an exterior domain, i.e. $\mathbb{R}^N\setminusΩ$ bounded, or the whole of $\mathbb{R}^N$. The potential $a\in L^{N/2}_{\rm loc}(\mathbb{R}^N)$ is assumed to be strictly positive and such that there exists $\lim_{|x|\to\infty}a(x):=a_\infty$, with $a_\infty>0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge …
We consider the focusing cubic NLS in the exterior $Ω$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)<E(Q)M(Q)$ and $\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$, the corresponding solution to the initial-value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, $Q(x)$ denotes the ground state for the focusing cubi…
We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surpris…
In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation} \label{eq_abst_1}-Δu = (1+g(x)) | \nabla u|^p \qquad \mbox{ in } B_1, \qquad u = 0 \mbox{ on } \;\; \partial B_1, \qquad \mbox{ and} \end{equation} \begin{equation} \label{eq_abst_2} -Δu = | \nabla u|^p \qquad \mbox{ in } Ω, \qquad u = 0 \mbox{ on } \;\; \partial Ω. \end{equation} In the first problem $B_1$ is the unit ball in $ \mathbb{R}^N$ and in the second $Ω$ i…
In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.
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.
This is the second and last paper of a two-part series in which we prove the $C^2$-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. In the first paper (arXiv:1702.05715), we showed that the maximal globally hyperbolic future development of an admissible asymptotically flat Cauchy initial data set is $C^2$-future-inextendible provided that an $L^2$-averaged (inverse) polynomial lower bound for the derivative of the scalar field holds along each horizon. In this paper, we show …
Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains
本文研究外部区域中的半线性椭圆方程，在权重满足特定对称与单调性条件下，证明了超临界指数范围内正解的存在性、非径向解的多重性及对称性破缺现象。
The Fujita phenomenon for nonlinear parabolic problems dtu = \deltau + up in an exterior domain of RN under the Robin boundary conditions is investigated in the superlinear case. As in the case of Dirichlet boundary conditions, it turns out that there exists a critical exponent p = 1+2/N such that blow-up of positive solutions always occurs for subcritical exponents, whereas in the supercritical case global existence can occur for small non-negative initial data.
The Fujita phenomenon for nonlinear parabolic problems $\partial tu = Δu + up$ in an exterior domain of RN under the dynamical boundary conditions is investigated in the superlinear case. As in the case of Dirichlet boundary conditions, it turns out that there exists a critical exponent $p = 1+2/N$ such that blow-up of positive solutions always occurs for subcritical exponents, whereas in the supercritical case global existence can occur for small non-negative initial data.
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.
In this paper, we study the global existence, uniqueness and large-time behavior of spherically symmetric solution of a viscous radiative and reactive gas in an unbounded domain exterior to the unit sphere in $\mathbb{R}^{n}$ for $n\geq 2$. The key point in the analysis is to deduce certain uniform estimates on the solutions, especially on the uniform positive lower and upper bounds on the specific volume and the temperature.
Existence of positive solutions for Kirchhoff type problems with critical exponent in exterior domains
该研究采用变分法，针对外区域中带临界指数的Kirchhoff型问题，通过建立全局紧引理、构造合适的极小极大值，在参数与孔洞适当小的条件下得到正解的存在性，补充了相关领域的研究空白并改进了部分已有结果。
This paper is devoted to prove the existence and nonexistence of positive solutions for a class of fractional Schrodinger equation in RN of the We apply a new methods to obtain the existence of positive solutions when f(u) is asymptotically linear with respect to u at infinity.
In this paper we construct nontrivial exterior domains $Ω\subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -Δu +u -u^p=0,\ u >0 & \mbox{in }\; Ω, {1mm] \ u= 0 & \mbox{on }\; \partial Ω, [1mm] \ \frac{\partial u}{\partial ν} = \mbox{cte} & \mbox{on }\; \partial Ω, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found…
We consider the Monge-Ampère equation $\det(D^2u)=f$ where $f$ is a positive function in $\mathbb R^n$ and $f=1+O(|x|^{-β})$ for some $β>2$ at infinity. If the equation is globally defined on $\mathbb R^n$ we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for $n\ge 3$ the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption $β>2$ is sharp for all the results in this article.
We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^θ\nabla u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question when $N':=N+θ>2$. In particular, for $N' > 2$ and $τ:=\ell-θ>-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2τ}{N'-2}$, while for any $ 0<p \leq p_s$, the only no…
In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison techniques, we show the existence, uniqueness, non-existence and regularity of the solutions. We also obtain a bifurcation-type result.
We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric sin…
We study existence and multiplicity of positive radial solutions for a coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the boundary conditions are nonlocal. We use a non-standard cone to establish existence of solutions by means of fixed point index theory.
We study the long time existence of solutions to nonlinear wave equations with power-type nonlinearity (of order $p$) and small data, on a large class of $(1+n)$-dimensional nonstationary asymptotically flat backgrounds, which include the Schwarzschild and Kerr black hole space-times. Under the assumption that uniform energy bounds and a weak form of local energy estimates hold forward in time, we give lower bounds of the lifespan when $n=3, 4$ and $p$ is not bigger than the critical one. The lower bounds for three dimensional subcritical and four dimensional critical cases are sharp in genera…
We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spec…
We study classical positive solutions of the biharmonic inequality $-Δ^2 v \geq f(v)$ in exterior domains in $\mathbb{R}^n$ where $f:(0,\infty)\to (0,\infty)$ is continuous function. We give lower bounds on the growth of $f(s)$ at $s=0$ and/or $s=\infty$ such that this inequality has no $C^4$ positive solution in any exterior domain of $\mathbb R^n$. Similar results were obtained by Armstrong and Sirakov [ Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011) 2011-2047] for $-Δv\ge f(v)$ using a method which depe…
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。
Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities
本文研究了具空腔区域中渐平坦变系数椭圆方程的解，通过基于加权能量的新单调性公式，推广了经典 Rellich $L^2$ 结果，证明了在更宽 $L^p$ 范围内解恒为零，确立了更尖锐的可积性阈值。
This paper is concerned with the spatial propagation of nonlocal dispersal equations with bistable or multistable nonlinearity in exterior domains. We obtain the existence and uniqueness of an entire solution which behaves like a planar wave front as time goes to negative infinity. In particular, some disturbances on the profile of the entire solution happen as the entire solution comes to the interior domain. But the disturbances disappear as the entire solution is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile and cont…
Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain
研究外域上含奇异Hardy-Rellich势、加权非线性和正源项的双调和热方程弱解的存在性与非存在性，识别两种不同临界情形，提出Fujita型临界指数及源项衰减指数的第二临界指数，扩展了Tobakhanov近期工作。
We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball $\mathbb{B}_{r_0}(0)\subset \mathbb{R}^n$ for $n\geq 3$ and some positive constant $r_0>0$, where the boundary data $v_-$ and the far field state $v_+$ of the initial data are prescribed and correspond to a stationary wave. It is shown in \cite{Hashimoto-Matsumura-JDE-2019} that a sufficient condition to guarantee the existence of such a stationary wave is $v_+<0, v_-\leq |v_+|+μ(n-1)/r_0$. Since the stationary wave is no longer monotonic, its nonline…
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。
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem.
Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$
本文研究二维星形障碍外部拟线性波动方程小数据解的长时间存在性，通过建立Morawetz型能量估计和加权L^2乘积估计，得到几乎尖锐的存在寿命下界。
In this note we obtain semiclassical resolvent estimates for non-trapping long range perturbations of the Laplacian on asymptotically Euclidean manifolds. Our proof is based on a positive commutator argument which differs from Mourre-type estimates by making the commutant also positive. The resolvent estimates, including the weighting of the Sobolev spaces in the estimates, are an immediate consequence.
In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for systems of equations involving potential terms and the fractional Laplacian in an exterior domain. Such problems arise in the analysis of a priori estimates of solutions. The class of problems we consider in this article is quite general compared to the literature. The main ingredient for our proofs is the hitting time estimates for the symmetric $α$-stable process and probabilistic representation of the super-solutions.
We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval $(-1,1)$. Our control function is localized in an open set $\mathcal O$ in the exterior of $(-1,1)$, that is, $\mathcal O\subset (\mathbb{R} \setminus (-1,1))$. We show that there exists a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the…
On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains
本文研究外部区域半线性热方程解关于不变测度的质量衰减，证明当且仅当 $1<p\leq\min\{2,1+2/N\}$ 时质量趋于零，否则解渐近自由，且 $N\geq3$ 时轮廓为高斯修正。
We solve the Jang equation with respect to asymptotically hyperbolic "hyperboloidal" initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.
On Jang's equation and the Positive Mass Theorem for asymptotically hyperbolic initial data sets with dimensions above three and below eight
本文在维度4至7的渐近双曲初始数据上求解Jang方程，为非旋量证明正质量定理提供了新途径，并推广了Sak21在三维情形下的结果。
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
We prove a positive mass theorem for $n$-dimensional asymptotically flat manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$ and the manifold is spin. This settles, for this class of manifolds, a question posed in a recent paper by the first author in connection with the long-term behavior of a certain Yamabe-type flow on scalar-flat compact manifolds with boundary.
Decay of weakly charged solutions for the spherically symmetric Maxwell-Charged-Scalar-Field equations on a Reissner-Nordström exterior space-time
该研究针对亚极端Reissner-Nordström外时空上的球对称Maxwell-带电标量场方程，在电荷足够小的条件下证明其解的有界性与衰减性，为相关黑洞稳定性研究奠定基础。
The Steklov problem for exterior domains: asymptotic behavior and applications
本文研究紧集外部修正Helmholtz方程的Steklov问题谱性质，获得了特征值和特征函数的小$p$渐近行为，并应用于随机过程理论揭示首达时间概率密度的长时间渐近行为，理论结果由有限元方法验证。
Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane
研究超平面外域中Robin本征值的优化问题，证明测地盘外的最低本征值在固定面积或周长约束下最大，且获得临界常数的反向不等式。
In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of…
Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains
本文为广义 $p$-Robin 问题的第一特征值建立了严格定量不等式，统一推导了有界区域中所有 $p$ 及小参数 $α$ 的精确渐近行为，并刻画了外区域特征值存在性与渐近性，改进了现有结果。
We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $Γ$ for the boundary of the obstacle, the relevant integral operators map $L^2(Γ)$ to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $Γ$ and are sharp up to factors of $\log k$ (where $k$ is the wavenumber), and the bounds on the norm of the inverse are valid for smooth $Γ$ and are observed to be sharp at least when $Γ$ is smooth with …
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes
本研究针对Kerr黑洞时空外部的轴对称线性扰动，构造了正定且规范不变的能量泛函，并证明该能量在时间上严格守恒，从而揭示了Kerr黑洞时空外部的动力学线性稳定性。
The Hyperboloidal and Spacetime Positive Mass Theorem in All Dimensions
本文基于Brendle-Wang的工作，证明了在任意维度中，对于渐近平坦和渐近双曲初始数据集的时空中正质量定理。
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