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Jingang Xiong

Publications and source records attributed to Jingang Xiong.

At least 19 recordsLinked to original sources
arXiv2026

Electrovacuum Black Hole Uniqueness

We prove the black hole uniqueness conjecture in the axially symmetric, stationary, electrovacuum setting, subject to the refined asymptotic analysis of the associated singular harmonic maps, which includes an analyticity hypothesis at the axes. More precisely, it is shown that any asymptotically flat solution of the Einstein--Maxwell equations in this class, with more than one black hole horizon component is either: Majumdar--Papapetrou, up to a duality rotation, in which case all logarithmic angle defects vanish, or every finite axis rod logarithmic angle defect is strictly negative and hence every interaction force is strictly attractive. The proof extends the singular harmonic map method used for vacuum Kerr uniqueness in [18].

gr-qc↗
arXiv2026

Kerr Black Hole Uniqueness

We prove the black hole uniqueness conjecture in the axially symmetric, stationary vacuum setting: there is no regular, asymptotically flat equilibrium configuration with more than one horizon component. More precisely, we establish that in every such multi-horizon configuration, the logarithmic angle defect along every bounded axis rod is negative, which implies that the net interaction force across each such segment is attractive. The proof is based on a refined asymptotic analysis of the associated singular harmonic maps, partially drawn from the companion work [16], together with a global maximum principle bound for the Weyl conformal factor arising from a novel scalar curvature related differential inequality.

gr-qc↗
arXiv2026

Łojasiewicz--Simon inequalities near bubbling configurations for the Yamabe functional on bounded domains

We establish Łojasiewicz--Simon type gradient inequalities for the Yamabe functional on bounded smooth domains near configurations consisting of finitely many concentrating bubbles, with or without a regular component. A weighted decomposition separates the finite-dimensional spectral and bubble parameters from an infinite-dimensional coercive remainder. We obtain quantitative estimates for the parameters, the remainder, and the corresponding energy gaps, in both the regular-plus-bubbling and the pure-bubbling regimes. These inequalities quantify the deviation from such surfaces and are essential for studying the dynamics of related parabolic flows.

math.AP↗
arXiv2026

Optimal regularity and fine asymptotics for very fast diffusion equations in bounded domains

We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that solutions belong to $C^{1,p+1}(\overlineΩ)$ in space for every positive time and are $C^\infty$ in time uniformly up to the boundary. Moreover, all their time derivatives belong to $C^{1,p+1}(\overlineΩ)$, and the exponent $p+1$ is optimal. These regularity estimates further yield fine long-time asymptotics toward the friendly giant solution, including a first-order expansion in the $C^{1,p+1}(\overlineΩ)$ topology and an improved convergence rate for the relative error in $C^{p+1}(\overlineΩ)$.

math.AP↗
arXiv2026

Asymptotic Analysis of Harmonic Maps With Prescribed Singularities

This is the first in a series of two papers to establish the mass-angular momentum inequality for multiple black holes. We study singular harmonic maps from domains of 3-dimensional Euclidean space to the hyperbolic plane having bounded hyperbolic distance to extreme Kerr harmonic maps. We prove that every such harmonic map admits a unique tangent harmonic map at the extreme black hole horizon. The possible tangent maps are classified and shown to be shifted `extreme Kerr' geodesics in the hyperbolic plane that depend on two parameters, one determined by angular momentum and another by conical singularities. In addition, rates of convergence to the tangent map are established. Similarly, expansions in the asymptotically flat end are presented. These results, together with those of Li-Tian [24,25] and Weinstein [35,36], provide a complete regularity theory for harmonic maps from $\mathbb R^3\setminus z\text{-axis}$ to $\mathbb H^2$ with these prescribed singularities. The analysis is additionally utilized to prove existence of the so called near horizon limit, and to compute the associated near horizon geometries of extreme black holes.

math.DG↗
arXiv2026

The Mass-Angular Momentum Inequality for Multiple Black Holes

This is the second in a series of two papers to address the conjectured mass-angular momentum inequality for multiple black holes. Our main result has two parts. In the first part, it is shown that either there is a counterexample to black hole uniqueness, in the form of a regular axisymmetric stationary vacuum spacetime with an asymptotically flat end and multiple degenerate horizons which is `ADM minimizing', or the following statement holds. Complete, simply connected, maximal initial data sets for the Einstein equations with multiple ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum, under the assumption of nonnegative energy density and axisymmetry. Moreover, equality is achieved in this mass lower bound only for a constant time slice of an extreme Kerr spacetime. Consequently, due to the nonexistence of stationary vacuum two-black hole configurations, the mass-angular momentum conjecture is verified for two black holes. In the second part, the mass-angular momentum inequality is established unconditionally, but with a suboptimal constant of $\frac{1}{2}$. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic map energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the harmonic map equations and their linearization. Additionally, we establish and utilize the Penrose inequality for manifolds with asymptotically cylindrical ends.

math.DG↗
arXiv2026

The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II

Continuing the work of the second author (2017), we study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. This paper addresses two open questions in this setting: the existence of Delaunay-type log-periodic solutions posed by del~Pino--Musso--Pacard (2007), and the asymptotic classification of singular solutions posed by Bidaut-Véron--Ponce--Véron (2007). We establish a global branch of positive log-periodic solutions along which blow-up occurs at a uniquely determined period. We also construct the corresponding concentrating family and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scale-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli--Gidas--Spruck (1989).

math.AP↗
arXiv2026

Optimal Weighted Smoothing and Asymptotics of Ancient Solutions for Fast Diffusion Equations

The Cauchy--Dirichlet problem for the fast diffusion equation on a smooth bounded domain admits a natural bound in the weighted space $L^p_{Φ_1}$, where $Φ_1$ is the first Dirichlet eigenfunction. A key regularization question is whether this implies the stronger $L^\infty$ bound. We provide a complete resolution, showing that the critical exponent coincides with the classical Brezis--Turner exponent from semilinear elliptic theory. As a primary application, we derive improved global Harnack inequalities and describe asymptotic behavior of positive ancient solutions.

math.AP↗
arXiv2026

Existence of non-radial entire solutions for the Hénon equation beyond even exponents

This paper is concerned with the existence of non-radial positive classical solutions for the critical Hénon equation \[ -Δu=|x|^αu^{\frac{N+2+2α}{N-2}} \qquad \text{in }\mathbb R^N, \] where \(α>0\) and \(N\ge3\), satisfying the Newtonian-type decay condition at infinity. Gladiali, Grossi and Neves (2013) proved existence for the discrete sequence $α_k=2(k-1)$, $k\in\mathbb N$, and conjectured that non-radial solutions may exist only at these special values. We disprove this conjecture by establishing existence for a continuum of exponents near each \(α_k\): for every even $k>\frac{N-2}{2}$, non-radial solutions persist for parameters \(α\) close to, and different from, \(α_k\). We recast the problem as a semilinear elliptic equation with Sobolev-supercritical exponent on the cylinder via the Emden--Fowler change of variables. Our argument is formulated directly on the cylindrical domain, thereby streamlining the characterization of the kernel of the linearized operator via Pöschl--Teller spectral theory, avoiding the ball-exhaustion technique employed in the original work, and allowing us to compute the bifurcation slope and verify the non-verticality condition.

math.AP↗
arXiv2026

Extremal Alexandrov estimates: singularities, obstacles, and stability

The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Ampère measure and yields, for two convex functions of $n$ variables with the same boundary values, a sup-norm bound with exponent $1/n$ in the measure discrepancy. We show that this exponent is not optimal in the small-discrepancy regime once one of the functions is non-degenerate in the sense of having Monge-Ampère density bounded above and below by two positive constants. We prove sharp quantitative estimates comparing two convex functions by the total variation of the difference of their Monge-Ampère measures: in dimensions $n\ge 3$ the optimal dependence is quadratic in the natural mass scale, while in dimension $n=2$ the optimal dependence contains a logarithmic correction. These rates are shown to be optimal for all small discrepancies. A key structural ingredient is a characterization of extremizers. We identify the pointwise minimizers and maximizers in the admissible class and prove that they are realized, respectively, by solutions to Monge-Ampère equations with an isolated singularity and by solutions to Monge-Ampère equations with a linear obstacle. This extremal description reduces the sharp estimates to a precise asymptotic analysis of these two model configurations. Assuming further that the domain and the non-degenerate reference function are $C^{2,α}$ and uniformly convex, we obtain sharp pointwise two-sided asymptotics at interior points with explicit leading constants. Finally, in dimensions $n\ge 3$ we establish a stability phenomenon: if the pointwise estimate is nearly saturated, then the measure discrepancy must concentrate near the point at the natural scale, quantifying rigidity of almost-extremal configurations.

math.AP↗
arXiv2026

Sharp global Alexandrov estimates and entire solutions of Monge-Ampère equations

This paper continues our work [19] on sharp Alexandrov estimates. We obtain a sharp global uniform distance estimate from a convex function to the class of unimodular convex quadratic polynomials in terms of the total variation of its Monge-Ampère defect measure relative to Lebesgue measure. The estimate has an explicit optimal constant, and the inequality is strict in the regime of positive finite defect mass. In this regime we further prove asymptotic rigidity at infinity: every such convex function admits a unique quadratic asymptote with an explicit convergence rate, and satisfies a sharp affine invariant global Alexandrov estimate with equality if and only if the function solves the isolated singularity problem or the hyperplane obstacle problem. Standard subsolution methods are not well suited to this measure-theoretic setting and typically do not yield sharp constants, while the sharp Alexandrov estimates developed in our earlier work [19] play a central role here. As an application, for entire solutions of Monge-Ampère equations with multiple (possibly infinitely many) isolated singularities, we give an explicit quantitative mass-separation condition ensuring strict convexity and hence smoothness away from the set of the isolated singularities.

math.AP↗
arXiv2025

On the singular set of the free boundary for a Monge-Ampère obstacle problem

This is a continuation of our earlier work [14] on the Monge-Ampère obstacle problem \[ \det D^2 v = v^q χ_{\{v>0\}}, \quad v \geq 0 \text{ convex} \] with $q \in [0,n)$, where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Ampère obstacle problem.

math.AP↗
arXiv2025

Regularity and classification of the free boundary for a Monge-Ampère obstacle problem

We study convex solutions to the Monge-Ampère obstacle problem \[ \operatorname{det} D^2 v=g v^qχ_{\{v>0\}}, \quad v \geq 0, \] where $q \in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the $L_p$ Minkowski problem. We establish $C^{1, α}$ regularity for the strictly convex part of the free boundary $\partial\{v=0\}$. Furthermore, when $g \in C^α$, we prove a Schauder-type estimate. As a consequence, when $g\equiv 1$, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets $\{v=0\}$. Combined with existing results, this provides a complete classification of entire solutions for the case $q=0$.

math.AP↗
arXiv2024

Extinction profiles for the Sobolev critical fast diffusion equation in bounded domains. I. One bubble dynamics

In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble.

math.AP↗
arXiv2024

Hölder regularity for the linearized porous medium equation in bounded domains

In this paper, we systematically study weak solutions of a linear singular or degenerate parabolic equation in a mixed divergence form and nondivergence form, which arises from the linearized fast diffusion equation and the linearized porous medium equation with the homogeneous Dirichlet boundary condition. We prove the Hölder regularity of their weak solutions.

math.AP↗