Aug 2026· 1 citation· ⚡ 1 influential· 27 references
Mathematics
Abstract
We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for \(3\)-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.
We prove strict convexity of the square-root transformation $v=-\sqrt{-u}$ for admissible solutions of a graphical $\sigma_2$-curvature Dirichlet problem on smooth uniformly convex domains. A key ingredient is a constant-rank theorem for $D^2v$, proved by a direct Ma-Xu type argument in dimension three and by the Bian-...
We establish a concavity inequality for the Hessian quotient operators $\frac{\sigma_k}{\sigma_l}$ in the cases $k-l\in\{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{\sigma_k...
We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description...
Fernando Charro, M. del Mar González, María Soria-Carro· 0 citations
We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radi...
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...
We prove that the gradient graph of any $W^{2,1}$ semi-convex viscosity solution to the three-dimensional subcritical special Lagrangian equation is area-minimizing and smooth. Furthermore, we prove the smoothness of $W^{2,1}\cap C^{1,1/3+}$ semi-convex viscosity solutions. This yields progress on the conjecture propos...
Zhen-Yu Fan· 0 citations
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