We prove that every entire smooth plurisubharmonic solution of the self-shrinking $J$-equation on $\mathbb{C}^n$ is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} as a special case. More generally, our method applies to a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the inverse complex Hessian quotient operators $-\sigma_{k-1}/\sigma_{k}$ for $1\leq k\leq n$.
We consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral $\int_{\Omega} g(|\nabla u|)\,dx$ with a strictly convex density $g: [0,\infty)\rightarrow \mathbb{R}$ being of linear growth. We show that the condition $\int_{0}^{\infty} t\,g''(...
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...
In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for con...
We derive sharp $\ell^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to\{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or disc...
Sung-Yi Liao, José Madrid, E. Palsson et al.· 0 citations
In this paper, on a bounded domain $\Omega\subset {\bf R}^n$, we consider a nonlocal problem of the type $$\cases{-\Delta u=Q(u,\lambda)f(u)&in $\Omega$ \cr&\cr u=0&on $\partial\Omega$\cr}$$ where $Q:H^1_0(\Omega)\times {\bf R}\to {\bf R}$, proving, under suitable assumptions, the existence of at least three weak solut...
We prove a third-order derivative estimate for convex solutions to the real Monge-Amp\`ere equation ${\rm det}\,{\rm Hess}(u) = 1$ on an open set in $\mathbb{R}^{2m}$ under the additional assumption that ${\rm Hess}(u)$ lies in ${\rm Sp}(2m,\mathbb{R})$ at every point. Our method is a geometric interpretation and exten...
Jose Rafael Santiago Arellano· 0 citations
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