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Bao Yu

Publications and source records attributed to Bao Yu.

7 recordsLinked to original sources
arXiv2026

Sharp Hessian integrability for the planar Monge--Ampère equation

We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying $$ λ\,dx\leqμ_u\leqλ^{-1}\,dx, \qquad 0<λ<1. $$ The distributional Hessian satisfies the sharp endpoint estimate $$ D^2u\in L_{\mathrm{loc}}^{P,\infty}, \qquad P=\frac{1+λ}{1-λ}. $$ This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent $P$. We further construct strictly convex $C^1$ solutions for which the distribution function of the Laplacian is comparable to $T^{-P}$ for all sufficiently large $T$. Hence both the exponent and the weak Lorentz endpoint are optimal: $L^{P,\infty}$ cannot be replaced by $L^{P,b}$ for any finite $b>0$. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.

math.AP↗
arXiv2026

Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation

We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. This resolves, in the classical strictly spacelike setting, the nonexistence conjecture of Byeon, Ikoma, Malchiodi, and Mari, including the critical endpoint. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Then a weighted trace-free tensor identity from the invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument and Souplet-type feedback inequality, yields a unified proof in the subcritical and critical ranges. The upper endpoint is sharp for $n\geqslant3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.

math.AP↗
arXiv2026

$C^{1,1}$ Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains

In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.

math.CV↗
arXiv2026

Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities

Let $N=m+1$ and consider the thin ellipsoid \[ Ω_\varepsilon=\{(y,x_N)\in\mathbb R^m\times\mathbb R:\ |y|^2+\varepsilon^{-2}x_N^2<1\}. \] We prove that in every sufficiently large dimension, for every sufficiently small $\varepsilon>0$, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity $f_\varepsilon$ for which the extremal solution in $Ω_\varepsilon$ is an unbounded $H^1_0$ solution. Combining this result with Dancer's thin-domain regularity theorem for the Gelfand nonlinearity $f(t)=e^t$, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis' Open Problem~6.1 is resolved in every sufficiently large dimension.

math.AP↗
arXiv2026

Sharp One-bubble Critical-Point Stability and Global Compactness for the Sobolev Trace Inequality

Let $n\ge3$ and $1<p<n$. We first prove the local trace analogue of the sharp one-bubble critical-point stability theorem of Liu and Zhang~\cite{LiuZhang2025}: near a positive trace-bubble, the Euler--Lagrange residual controls the gradient distance to the normalized trace-bubble manifold with the sharp power $\max\{1,p-1\}$. Then, we establish a Struwe-type compactness theorem for the critical trace functional, which gives the trace counterpart of the Mercuri--Willem decomposition~\cite{MercuriWillem2010}. Combining Struwe-type compactness with the local stability estimate yields a sharp quantitative one-bubble critical-point stability theorem.

math.AP↗
arXiv2026

Quantitative Stability for Minimizing Yamabe Metrics with minimal boundary

In this paper, we investigate the stability of minimizing Yamabe metrics on compact manifolds with boundary, in the sense introduced by Escobar. We show that if a function nearly minimizes the Yamabe energy, then the associated conformal metric is quantitatively close to a minimizing Yamabe metric within its conformal class. Moreover, this closeness is controlled by an appropriate power of the Yamabe energy deficit.

math.DG↗