We prove a Pogorelov interior estimate for strictly plurisubharmonic solutions of the complex Monge-Amp\`ere equation $\det(u_{i\bar{j}})=1$ with homogeneous Dirichlet data under the condition that, for some constant $\kappa\geq 1$, $(\kappa u_{i\bar{j}}-u_{is}u^{s\bar{t}}u_{\overline{tj}})$ is non-negative definite. For $\kappa=1$, this gives the estimate for real convex solutions. The estimate depends only on the dimension, $\kappa$, and the $C^0$ and $C^1$ norms of the solution. As an application, every smooth entire convex solution of $\det(u_{i\bar{j}})=1$ on $\mathbb{C}^n$ is a real quadratic polynomial. The same conclusion holds if real convexity is assumed only outside a compact subset of $\mathbb{C}^n$.
Consider \begin{equation*}\ \left\{ \begin{aligned} -\mathcal_{\lambda,\Lambda}^{+}(D^{2}u)&=\mu u+a(x)u^{p}\quad\text{in}~\Omega,\\ u&=0\quad\text{on}~~\partial\Omega, \end{aligned} \right. \end{equation*} where $ \ Omega$ is a smooth bounded domain, $a$ is a continuous function which is allowed to change sign. Moreov...
Karan Rathore, Mohan Mallick, Ram Baran Verma· 0 citations
Let $n\ge2$ and let $\Omega\subset\mathbb R^n$ be an arbitrary bounded open convex set. The author prove that, for $p>n$, the Dirichlet problem \[ \det D^2u=(-u)^p\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega, \qquad u>0\quad\text{in }\Omega \] has at most one convex Alexandrov solution. The proof is b...
We consider entire solutions $u$: $\mathbb{R}^n \rightarrow \mathbb{R}$, $n \ge 2$, of the equation $\operatorname{div}(\nabla f(\nabla u)) = 0$ with convex density $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$ given for example by $f(\nabla u) = \sum_{i=1}^n |\partial_i u|^{p_i}$ with exponents $p_i \ge 2$ including Giaq...
We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying $$ \lambda\,dx\leq\mu_u\leq\lambda^{-1}\,dx, \qquad 0<\lambda<1. $$ The distributional Hessian satisfies the sharp endpoint estimate $$ D^2u\in L_{\mathrm{loc}}^{P,\infty}, \qquad P=\frac{1+\lambda}{1-\lambda}....
Xiao-Rui Lu, Jin Yan, Bao-Yue Yu et al.· 0 citations
We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1<p<p_...
Nirenberg's theorem gives interior $C^{2,\alpha}$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^2u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vl\u{a}du\c{t} shows that such solutions need not be $C^2$. We construct a famil...
Bao-Zhi Chu· 0 citations
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