YanYan Li, Luc Nguyen, Bo Wang
We consider the problem of prescribing the $σ_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for $n = 4$ and $k = 2$.
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Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_σ(v)= Kv^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $\mathbb{S}^n$ for all $σ\in (0,n/2)$, where $P_σ$ is the intertwining operator of order $2σ$. Finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures. When $σ=1$, it is the prescribing scalar curvature problem or the Nirenberg problem,…
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on $§^n$ for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory inte…
In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical sol…
In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $σ=1/2,$ when the prescribing $σ$-curvature function satisfies the $(n-2σ)$-flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li in \cite[CPAM…
In this paper we study the problem, posed by Troyanov, of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function $K$ changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, by a min-max approach and a new compactness argument we give sufficient conditions on $K$, concerning…
Smooth solutions to the Christoffel problem in $\mathbb{H}^{n+1}$
本文通过证明满秩定理建立双曲空间中 Christoffel 问题光滑解的存在性，并据此推出 $\mathbb{S}^n$ 上给定标量曲率的 Nirenberg-Kazdan-Warner 问题解的存在性。
We study the problem of prescribing $σ_k$-curvature for a conformal metric on the standard sphere $\mathbb{S}^n$ with $2 \leq k < n/2$ and $n \geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $β\in [2,n)$. We prove among other things the following statements. (1) When $β>n-2k$, the solution set is compact, has a nonzero total de…
A Machine Learning Approach to the Nirenberg Problem
本文提出Nirenberg神经网络，通过机器学习方法解决高斯曲率指定问题，实现低损失训练并区分可实现与不可实现曲率。
We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan-Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.
In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^2} f>0$ at the saddle points.
This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.
We consider the fractional Nirenberg problem on the standard sphere $\mathbb{S}^n$ with $n\geq 4$. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.
We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.
In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for some nonlinear partial differential systems.
Prescribing curvatures on surfaces with conical singularities and corners
本文研究了带锥奇点和角点的带边界紧致曲面上的高斯曲率与测地曲率预定问题，通过将新变分公式推广至奇异情形求得解，是该奇异曲面预定曲率问题的首个结果。
We prove in this article that the local image of each conformal $Q$-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.
Admissibility, Topology, and the $σ_k$-Loewner--Nirenberg problem
研究$σ_k$-Loewner--Nirenberg问题中可容许性条件的拓扑障碍，通过例子说明存在拓扑限制。
In this paper, the fractional order curvature equation $(-Δ)^γu = (1 + \varepsilon K(x))u^{\frac{N + 2γ}{N - 2γ}}$ in $\mathbb{R}^N$ is considered. Assuming $K(x)$ has two critical points satisfying certain local conditions, we prove the existence of two-peak solutions.
We establish theorems on the existence and compactness of solutions to the $σ_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order der…
We introduce a flow approach to the generalized Loewner-Nirenberg problem $(1.5)-(1.7)$ of the $σ_k$-Ricci equation on a compact manifold $(M^n,g)$ with boundary. We prove that for initial data $u_0\in C^{4,α}(M)$ which is a subsolution to the $σ_k$-Ricci equation $(1.5)$, the Cauchy-Dirichlet problem $(3.1)-(3.3)$ has a unique solution $u$ which converges in $C^4_{loc}(M^{\circ})$ to the solution $u_{\infty}$ of the problem $(1.5)-(1.7)$, as $t\to\infty$.
In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove the existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality…
We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,γ}_{\rm loc}$ for any $γ> \frac{1}{k}$. Optimal regularity for solutions to the $σ_k$-Yamabe problem on annuli with finite constant boundary values is also established.
This paper focuses on the problem of prescribing mean curvature on the unit ball. Assume that $f$, which is allowed to change sign, satisfies Morse index counting or certain kind of symmetry condition. By using a negative gradient flow method, we then prove that $f$ can be realized as the boundary mean curvature of some conformal metric.
In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.
We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partialΩ$ is a smooth bounded hypersurface (of codimension one). When $\partialΩ$ contains a compact smooth submanifold $Σ$ of higher codimension with $\partialΩ\setminusΣ$ being compact, we also give a `sharp' condition for the di…
We consider the multi-bump solutions of the following fractional Nirenberg problem \begin{equation}\label{01} (-Δ)^s u=K(x)u^{\frac{n+2s}{n-2s}}, \;\;\;\;u>0\;\;\text{ in }\mathbb{R}^n, \end{equation} where $s\in (0,1)$ and $n>2+2s$. If $K$ is a periodic function in some $k$ variables with $1\leq k<\frac{n-2s}2$, we proved that \eqref{01} has multi-bump solutions with bumps clustered on some lattice points in $\mathbb{R}^k$ via Lyapunov-Schmidt reduction. It is also established that the equation \eqref{01} has an infinite-many-bump solutions with bumps clustered on some lattice points in $\mat…
Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon
本文研究高维球面上Nirenberg问题次临界逼近中带残余质量的爆破现象，刻画爆破点与速率，计算拓扑贡献，并在弱极限非退化时构造爆破解。
The horospherical $p$-Christoffel-Minkowski problem in hyperbolic space
本文研究双曲空间中的horospherical $p$-Christoffel-Minkowski问题，通过粘性方法证明满秩定理，建立了一致$h$-凸解的存在性，并蕴含了共形几何中Nirenberg型问题解的存在性。
This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$ \{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}. $$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $ φ$ is globally Lipschitz, or $n \leq 8$ and $ \frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constr…
We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
Prescribing nearly constant curvatures on balls
针对 Kazdan-Warner 问题的边界情形，研究通过共形度量变换在圆盘及高维球上预设近常的内、外曲率，并利用 Ljapunov-Schmidt 方法证明负内曲率、正边界曲率条件下的新存在性结果。
We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures $K$ contained in naturally defined stable regions. We prove that in such stable regions, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also p…
Prescribing Chern scalar curvatures on noncompact manifolds
针对非紧非 Kähler 流形上的预定 Chern 标量曲率问题，论文借助新的解析条件建立 Kazdan-Warner 型方程的存在性结果，并据此给出 Ni 经典多重性定理的新证明。
A note on borderline Brezis-Nirenberg type problems
该研究聚焦散度型椭圆算子的Brezis-Nirenberg问题，分析边界上椭圆性点对该问题的影响，为相关边界问题研究提供理论参考。
We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem.
In this paper, we study the decomposition of Nehari manifold for the Brézis-Nirenberg problem with nonhomogeneous Dirichlet boundary conditions. By using this result, the Lusternik-Schnirelman category and the minimax principle, we establish a multiple result (four solutions) for the Brézis{Nirenberg problem with nonhomogeneous Dirichlet boundary conditions.
An orbit-like foliation is a singular foliation on a complete Riemannian manifold $M$ whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of $M$ into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of or…
Prescribing almost constant curvatures on manifolds with boundary
本文研究标准球面上预定数量曲率与边界平均曲率的边界情形，利用微扰方法和 Han-Li ansatz 证明了当预定曲率接近常数时共形度量的存在性。
In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \e…
We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that blows up in the region where the prescribed functions are negative. This is a different phenomenon from the classical Nirenberg problem.
In the present paper, we study the effect of an elliptic operator in divergence form L defined in a bounded open set on the existence and nonexistence of positive solutions of the associated nonlocal Brezis-Nirenberg problem involving powers of L.
In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions.
In this paper, we study the Brezis-Nirenberg problem on bounded smooth domains of R3. Using the algebraic topological argument of Bahri-Coron[2] as implemented in [6] combined with the Brendle[4]- Schoen[8]'s bubble construction, we solve the problem for non-contractible domains.
This paper is a continuation of our earlier work "[T. Jin, Y.Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, to appear in J. Eur. Math. Soc.]", where compactness results were given on a fractional Nirenberg problem. We prove two existence results stated there. We also obtain a fractional Aubin inequality.
The main objective of this paper is to establish the convergence for the fractional $p$-Laplacian of nonnegative sequence of functions with $p>2$. Further, we show the blow-up phenomena for solutions to the extended Nirenberg problem modeled by fractional $p$-Laplacian with the prescribed negative functions.
Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon-Douglis-Nirenberg are investigated. In the whole- and half-space case we construct an explicit formula for the solution and establish coercive Lp estimates. The estimates generalize a famous result of Agmon, Douglis and Nirenberg for elliptic problems to the time-periodic case.
Singular Solutions of the Loewner-Nirenberg Problem in Conic Domains with Prescribed Singularity at Vertices
本文针对锥形区域上的Loewner-Nirenberg问题开展研究，证明了顶点附近带指定任意高阶精度渐近展开的正奇性解存在。
We study the asymptotic behaviors of solutions of the Loewner-Nirenberg problem in singular domains and prove that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary. The conformal structure of the underlying equation plays an essential role in the derivation of the optimal estimates.
We show that the classical Brezis-Nirenberg problem $$ -Δu=u|u| + λu\ \hbox{in}\ Ω, u=0\ \hbox{on}\ \partialΩ, $$ when $Ω$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $Ω$ as $λ$ approaches a suitable value $λ_0>0.$
We prove some existence and nonexistence results for a class of critical $(p,q)$-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.
On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting
本文利用Morse理论和计数指标方法，在无对称性假设下证明标准球面上Nirenberg问题在扰动背景下存在任意多个共形度量解。
We consider the classical Brezis-Nirenberg problem in the unit ball of $\mathbb{R}^N$, $N\geq 3$ and analyze the asymptotic behavior of nodal radial solutions in the low dimensions $N=3,4,5,6$ as the parameter converges to some limit value which naturally arises from the study of the associated ordinary differential equation.
We develop a new method for estimating the region of the spectral parameter of a generalized Brezis--Nirenberg problem for which there are no, non trivial, smooth solutions. This new method combines the standard Rellich--Pohozaev argument with a Hardy type inequality for bounded domains. The estimates we get are better than the usual estimates for low dimensions.
We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial solutions in the subcritical regime. We also show a non-existence result in star-shaped domains when the exponent is supercritical.
Prescribing $Q$-curvature on even-dimensional manifolds with conical singularities
该研究针对带锥形奇点的$2m$维闭流形的指定$Q$-曲率问题，通过爆破分析与极小极大变分论证，得到超临界情形下的存在性与多重性结果，是$m>1$时非球面超临界锥形流形的首个相关存在性结论。
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Key step is the study of the eigenvalues growth and eigenfunctions estimates.
For $p\in (1,2]$ and a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ let $μ_p(\barΩ,\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\barΩ$ of $Ω$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $μ_p$ on a planar convex domain: "Given a finite, nonnegative, Borel measure $μ$ on $\mathbb S^1$, find a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ such that $dμ_p(\barΩ,\cdot)=dμ(\cdot)$" is solvable if and only if $μ$ has centroid at the origin and its support $\mathrm{supp}(μ)$ does not comprise any pair of …
In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} w…
On Brezis-Nirenberg problems: open questions and new results in dimension six
本文研究Brezis-Nirenberg问题，回顾其历史，提供六维中的新结果，并列出开放问题。
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