Searcharxiv⌕ Search

arXiv subjects

Bao Yu

Publications and source records attributed to Bao Yu.

5 recordsLinked to original sources
arXiv2026

Singular Extremal Solutions on Thin Ellipsoids with Varying Nonlinearities

Let $N=m+1$ and consider the thin ellipsoid \[ \Omega_\varepsilon=\{(y,x_N)\in\mathbb R^m\times\mathbb R:\ |y|^2+\varepsilon^{-2}x_N^2<1\}. \] 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 $\Omega_\varepsilon$ 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, Br\'ezis' Open Problem~6.1 is resolved in every sufficiently large dimension.

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 $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key independent ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Thus pointwise strict spacelikeness automatically improves to a uniform spacelike gap, including in the critical and supercritical regimes. The proof combines a geometric Bernstein estimate, comparison with an explicit hyperbolic cap, and weighted trace-free tensor identities. The result extends the known radial nonexistence theorem to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.

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↗