Jonah A. J. Duncan, Luc Nguyen
We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $σ_k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.
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We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also s…
We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant $Y\leq 0$. Previous work by the second and third named authors \cite{ChenWang} showed that while the Yamabe flow always converges in a global weighted sense when $Y>0$, the flow must diverge when $Y\leq 0$. We show here in the $Y\leq 0$ case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.
Liouville-type results for the CR Yamabe equation in the Heisenberg group
本文在 Heisenberg 群中研究 CR Yamabe 方程，通过积分估计和散度公式，在无穷远逐点或积分条件下获得 Liouville 型结果，推广了 Jerison 和 Lee 的结论。
On the fully nonlinear Yamabe problem with constant boundary mean curvature. I
本文研究了完全非线性Yamabe问题，证明了共形不变二阶椭圆方程在半欧氏空间中的最优Liouville型定理。
Fixed-spinor radial solutions of a perturbed Euclidean spinorial Yamabe equation
针对受扰欧几里得旋量Yamabe方程，通过径向旋量空间的有限维约化，结合K在原点的行为与加权积分量，得到固定旋量径向解的存在及多重性准则。
Euclidean Domains with Nearly Maximal Yamabe Quotient
本文证明欧几里得域若其Yamabe商几乎达到球的最大值，则该域几乎为球，并给出与拟共形系数的定性比较。
A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundary
研究带内边界的渐近欧几里得流形中预定非正标量曲率与边界平均曲率的共形度量存在性，给出由目标曲率零集共形不变量刻画的充要条件，并据此建立 Yamabe 分类。
In this paper, we analyze the asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations on an asymptotically hyperbolic Riemannian manifold. We prove that Palais-Smale sequences can be decomposed into the solution of the limit equation plus a finite number of bubbles, which are the rescaling of the fundamental solutions to the fractional Yamabe equation on Euclidean space. We also verify the non-interfering fact for multi-bubbles.
We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also established via the method of moving planes. This bound increases the energy range for which Palais-Smale sequences of related variational problem has a non-trivial weak limit.
Asymptotic approach to singular solutions for the CR Yamabe equation
研究海森堡群中福兰德 - 斯坦 - 索伯列夫嵌入次临界逼近最优函数紧致性损失，防边界爆破时中心对称最大化器集中于中心，获远离极点爆破率与渐近轮廓，建立近临界索伯列夫问题对应渐近。
Asymptotic behavior of solutions to the Yamabe equation with an asymptotically flat metric
该研究证明了满足平坦阶数条件的渐近平坦流形上Yamabe方程正解的无穷远渐近二择一性，通过孤立奇点附近渐近分析推广了多类前人结果。
Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n \ge 7$ and of positive Yamabe type. If $ξ_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $ k \ge 1$ and $\varepsilon >0$, of a positive blowing-up solution $u_{\varepsilon}$ of $$\triangle_g u_{\varepsilon} +\big( c_n S_g +\varepsilon h\big) u_{\varepsilon} = u_{\varepsilon}^{2^*-1},$$ that blows up like the superposition of $k$ positive bubbles concentrating at different speeds at $ξ_0$. The method of proof combines a finite-dimensional reduction with the sharp…
Let $G=(V,E)$ be a locally finite graph, $Ω\subset V$ be a bounded domain, $Δ$ be the usual graph Laplacian, and $λ_1(Ω)$ be the first eigenvalue of $-Δ$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $α<λ_1(Ω)$, then for any $p>2$, there exists a positive solution to $-Δu-αu=|u|^{p-2}u$ in $Ω^\circ$, $u=0$ on $\partialΩ$, where $Ω^\circ$ and $\partialΩ$ denote the interior and the boundary of $Ω$ respectively. Also we consider similar problems involving the $p$-Laplacian and poly-Laplacian by the same method. Such p…
We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger-Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n \ge 3$. This is a generalisation of the so-called $C^0$-theory for positive solutions of Schrödinger-Yamabe type equations. To deal with the sign-changing case we develop a method of proof that combines an \emph{a priori} bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp \emph{a priori} blow-up estimates for a linear problem.
Classification of positive entire solutions of the CR Yamabe equation on the Heisenberg group
该研究完成了海森堡群上临界CR Yamabe方程正整体解的维度分类，通过利用Green表示的推论、共形协变性等，在无能量界的类中隔离出气泡流形，解决了欧氏方法无法适用的问题。
We obtain local pointwise second derivative estimates for $W^{2,p}$-strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry.
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data s…
Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (λ, λ+ 1] has a universal scaling limit as λ goes to infinity (depending only on the dimension of M). Our results also imply that if M has no conjugate points, then immersions of M into Euclidean space by an orthono…
We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case p=2. In this article we establish a priori estimates and regularity results for the $p$-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.
We investigate the dispersive properties of solutions to the Schrödinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrödinger flow on each eigenspace of the link manifold satisfies a weighted $L^1\to L^\infty$ dispersive estimate. In odd dimensions, the decay rate we compute is consistent with that of the Schrödinger equation in a Euclidean space of the same dimension, but the spatial weights reflect the more complicated regularity issues in frequency that we face in the form of the spectral m…
By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denote…
Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces
本文研究了热方程和海森方程在调制空间中初始数据的点收敛性，首次在加权调制空间中证明了热半群的点收敛性，并探讨了Hardy-Littlewood极大算子在调制空间中的作用。
Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces
本文研究 Damek-Ricci 空间上径向初值薛定谔方程的 Carleson 问题，证明了解逐点收敛的最优正则性指标为 $\alpha \ge 1/4$，与欧几里得情形一致。
Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem
本文证明在光滑紧致域上，若一个黎曼度量与欧几里得度量具有相同的边界度量和Dirichlet-to-Neumann映射且足够接近，则它们通过固定边界的微分同胚等距，即欧几里得度量具有局部刚性。
As a new technique it is shown how general pseudo-differential operators can be estimated at arbitrary points in Euclidean space when acting on functions $u$ with compact spectra. The estimate is a factorisation inequality, in which one factor is the Peetre--Fefferman--Stein maximal function of $u$, whilst the other is a symbol factor carrying the whole information on the symbol. The symbol factor is estimated in terms of the spectral radius of $u$, so that the framework is well suited for Littlewood--Paley analysis. It is also shown how it gives easy access to results on polynomial bounds and…
The anisotropic Calderón problem: rigidity near the Euclidean metric
该文证明三维及以上区域中各向异性Calderón问题在欧几里得度量附近的刚性：若光滑度量充分接近欧几里得度量且具有相同Dirichlet-to-Neumann映射，则必相差一个固定边界的微分同胚，方法结合调和坐标与Carleman估计。
In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.1. T…
Pointwise convergence of solutions of the Schrödinger equation along general curves on Damek-Ricci spaces
研究Damek-Ricci空间中薛定谔方程解沿一般曲线的逐点收敛问题，通过双曲空间反例排除宽逼近区域，并对满足Hölder与双Lipschitz条件的曲线建立初值正则性的临界界。
In this paper we prove global well-posedness of the critical surface quasigeostrophic equation on the two dimensional sphere building on some earlier work of the authors. The proof relies on an improving of the previously known pointwise inequality for fractional laplacians as in the work of Constantin and Vicol for the euclidean setting.
We prove extensions of the estimates of Aleksandrov and Bakel$'$man for linear elliptic operators in Euclidean space $\Bbb{R}^{\it n}$ to inhomogeneous terms in $L^q$ spaces for $q < n$. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and $L^2$ estimates.
Regularity and pointwise convergence for dispersive equations on $\mathbb{H}^2$
本文在二维实双曲空间上研究色散方程的 Carleson 问题，证明初始数据 Sobolev 正则性阈值 β≥1/2 即可保证解几乎处处逐点收敛，并将该结果推广到分数阶薛定谔方程、Boussinesq 方程和 Beam 方程，改进了已有最佳结果。
Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
研究 Damek-Ricci 空间上具渐近凹相位的色散方程的 Carleson 问题，获得解向径向初始数据几乎处处逐点收敛的几乎尖锐正则性阈值 $β>a/4$，该结果推广了经典欧氏结论。
Pointwise convergence for single-particle Schrödinger equation and Schrödinger equation with infinitely many particles on the torus and waveguide manifolds
本文研究环面和波导流形上单粒子及无穷多粒子薛定谔方程的点态收敛性，建立了时间极大Strichartz估计，给出了收敛的充分与必要条件，并推广至费米子系统。
In this paper we consider the Laplace-Beltrami operator Δon Damek-Ricci spaces and derive pointwise estimates for the kernel of exp(τΔ), when τ\in C* with Re(τ) \geq 0. When τ\in iR*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schrödinger equation associated …
Maximal estimates and pointwise convergence for solutions of certain dispersive equations with radial initial data on Damek-Ricci spaces
本文研究Damek-Ricci空间上带径向初值的三类色散方程的Carleson问题，通过刻画局部极大函数映射性质得到逐点收敛的精确界，还建立了色散方程的抽象转移原理用于证明主结果。
We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u) = w(|x|)g(u)$ on the Euclidean space $R^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behavior at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behavior at infinity of the growth rate function of $w$.
Logarithmic gradient estimate and Universal bounds for semilinear elliptic equations revisited
该研究利用Bernstein方法对满足次临界条件的半线性椭圆方程给出对数梯度与普适有界性估计，建立了曲率有下界流形上基本估计的关联，证明非负曲率下估计的κ-等价性，并得到欧氏空间方程的最优估计与Liouville定理。
A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set.
On the three dimensional Euclidean space, for data with finite energy, it is well-known that the Maxwell-Klein-Gordon equations admit global solutions. However, the asymptotic behaviours of the solutions for the data with non-vanishing charge and arbitrary large size are unknown. It is conjectured that the solutions disperse as linear waves and enjoy the so-called peeling properties for pointwise estimates. We provide a gauge independent proof of the conjecture.
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