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Shujun Shi

Publications and source records attributed to Shujun Shi.

7 recordsLinked to original sources
arXiv2026

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

We study nonnegative entire spacelike graphs in Lorentz--Minkowski space satisfying $σ_k(A[u])=u^p$, with $h_{ij}=-Wu_{ij}$ and $W=(1-|Du|^2)^{-1/2}$. For $2\leq k 2k$ and $k\leq p<k(n+2)/(n-2k)$, we prove that every such solution vanishes identically, without symmetry, decay, integrability, or curvature pinching assumptions. The integral proof combines a quantitative Newton inequality with a weighted divergence identity and a finite sequence of integrations by parts, using the angle variable $2(W-1)$. We also obtain a corresponding rigidity result for complete spacelike immersions. At the upper endpoint, we identify the loss of quadratic gradient coercivity; our argument does not settle the critical Liouville problem.

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↗