Searcharxiv⌕ Search

arXiv subjects

Shujun Shi

Publications and source records attributed to Shujun Shi.

9 recordsLinked to original sources
arXiv2026

Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems

Let $Ω\subset\mathbb R^n$, $n\ge3$, be a bounded uniformly convex $C^{3,γ}$ domain, where $0<γ<1$. We study the limit as the positive Robin parameter $α\to\infty$. For the $2$-Hessian torsion problem, let $U$ be the admissible Dirichlet solution. For all sufficiently large $α$, we obtain a unique admissible Robin solution $u_α$ and establish the optimal-order estimate \[ \|u_α-U\|_{C^{2,γ}(\overlineΩ)} \le Cα^{-1}. \] Moreover, $-\sqrt{-u_α}$ has a uniform positive lower bound on its Hessian, and hence $\sqrt{-u_α}$ is strictly concave. For the normalized eigenvalue problem, let $(λ_D,φ_D)$ be the normalized negative admissible Dirichlet eigenpair, and assume $φ_D\in C^{2,γ}(\overlineΩ)$. For all sufficiently large $α$, there is a unique normalized negative admissible Robin eigenpair $(λ_α,φ_α)$, which satisfies \[ \|φ_α-φ_D\|_{C^{2,γ}(\overlineΩ)} +|λ_α-λ_D|\le Cα^{-1}. \] The Hessian of $-\log(-φ_α)$ also has a uniform positive lower bound. The main analytic tool is a uniform Hölder estimate for linear Robin problems, with an $α^{-1}$ gain for $C^{2,γ}$ boundary data. A normalized augmented inverse removes the scaling kernel in the eigenvalue problem. A Schur-complement argument combines the $C^2$ convergence with boundary geometry to prove convexity up to the boundary and anisotropic Hessian lower bounds at the scale ${\rm dist}(x,\partialΩ)+α^{-1}$. The convergence rate $O(α^{-1})$ for the Robin solution functions is optimal even in $C^0$, as follows from their boundary conditions.

math.AP↗
arXiv2026

The Asymptotic Plateau Problem for $p$-Convex Hypersurfaces in Hyperbolic Space

We solve the asymptotic Plateau problem in hyperbolic space for the curvature given by the normalized geometric mean of the $p$-fold sums of the principal curvatures. For every $n\ge3$, $2\le p\le n-1$, and $σ\in(0,1)$, each bounded domain $Ω\subset\mathbb{R}^n$ with smooth mean-convex boundary admits a unique complete admissible vertical graph with prescribed curvature $σ$ and asymptotic boundary $\partialΩ\times\{0\}$. The result extends the known cases $(n,p)=(3,2)$ and $(4,3)$ to the stated range. The key curvature estimate is uniform with respect to the positive boundary height in the approximating Dirichlet problems. The proof establishes two lower bounds for the full third-derivative quadratic form under the linearized constraints: a sharp universal bound with coefficient $2/p$ and a stronger direction-dependent bound when the relevant principal curvature lies in a specified range.

math.DG↗
arXiv2026

Universal Spacelikeness Estimates and Liouville Rigidity for Lorentzian $σ_k$ Curvature Equations

We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $σ_k(A[u])=u^p$. For $2\le k<n$ and $p\ge k$, pointwise strict spacelikeness and closed $k$-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full Gårding cone, a Lorentzian cutoff, and hyperbolic-cap comparison. For $2\le k<n/2$ and $k\le p\le k(n+2)/(n-2k)$, every such solution vanishes, without symmetry, decay, finite-energy, curvature-pinching, or global Hessian assumptions. In the subcritical range we use a trace-free Newton tensor, a weighted divergence identity, and a finite descent. At the critical endpoint, the divergence identity retains an exact nonnegative defect and a positive quartic term. Direct estimates cover $2k<n\le4k+2$. For $n\ge4k+3$, we use compactness, concentration of the $k$-Hessian measure at the first crossing, a single-pole Pohozaev argument, core counting, a recursive defect estimate, and Souplet-type radius selection. A geometric corollary gives a rigidity result for complete spacelike immersions under the corresponding curvature equation.

math.AP↗
arXiv2026

Lewy-Type Nondegeneracy for Gradient-Dependent Elliptic Equations and Global Gradient Diffeomorphisms on Convex Rings

We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $Ω\subset\mathbb{R}^3$ and let $u$ solve \[ a^{ij}(Du)u_{ij}=0, \] where $a$ is a smooth, symmetric, positive definite matrix field on an open set containing $Du(Ω)$. We show that if the gradient map $Du$ is a local homeomorphism, then $\det D^2u$ never vanishes; hence $Du$ is a local $C^\infty$-diffeomorphism. Second, in every dimension, we consider capacitary solutions on convex rings. If the solution has no critical points and its level hypersurfaces are strictly convex, then ellipticity alone forces the Hessian to have one positive and $n-1$ negative eigenvalues, that is, inertia $(1,n-1)$. Consequently, the gradient is a global diffeomorphism onto a radially parametrized ring in gradient space. Both results apply to the $p$-Laplace and minimal surface equations. For the global minimal-surface result, existence of a smooth solution is assumed. Classical convex-ring results supply the noncriticality and strict level-set convexity needed in the global corollaries. Since the two coefficient matrices are real analytic on the relevant gradient ranges, the corresponding local and global gradient diffeomorphisms are real analytic.

math.AP↗
arXiv2026

Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains

Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.

math.AP↗
arXiv2025

On the solvability for a p-k-Hessian inequality

In this paper, we discuss the solvability of a p-k-Hessian entire inequality. We prove that the inequality with sub-lower-critical exponent admits no negative solutions. Moreover, the exponent is sharp. The proof is based on choosing suitable test functions and integrating by parts.

math.AP↗