In this paper, we study entire solutions and periodic correctors for the coupled-drift Monge-Amp\`ere equation \[ \det D^2u = \exp\{-a\cdot Du+b\cdot x+V(x)-c_0\}, \quad D^2u>0. \] For $V\equiv0$, we obtain a sharp classification of the whole-space solvability regimes: all entire smooth strictly convex solutions are quadratic when $a=b=0$; no such solution exists when $a\neq0$ and $a\cdot b\le0$; and non-quadratic entire solutions exist when $a=0$ and $b\neq0$, or when $a\cdot b>0$. For the null case $a\neq0$, $a\cdot b=0$, we give a scalar maximum-principle argument in every dimension $n\ge2$. For periodic $V$, we prove existence and uniqueness of the normalized pair $(\psi_A,c_A)$ solving the drifted cell problem \[ \det(A+D^2\psi) = \exp\{-a\cdot D\psi+V-c_A\}, \quad A+D^2\psi>0 \quad\text{on }\mathbb T^n. \] We also prove that any asymptotically quadratic entire solution must satisfy $b=Aa$. If its remainder is bounded, then the solution is the corresponding quadratic-periodic corrector up to an additive constant.
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 study positive solutions of the partially confined stationary nonlinear Schr\"odinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-\Delta + |y|^2$ is given by $[d,\infty)$. We prove that $\lambda\geq-d$ is necessary for po...
Zhijie Chen, Youjun Wang, Jian-Jun Zhang et al.· 0 citations
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
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. F...
We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^2 u=M|u|^p\nu$ with zero boundary values on bounded convex domains in $\mathbb{R}^n (n \geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $\nu$ is a locally finite Bor...
Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\g...
M. del Pino, Ignacio A. Guerra, M. Musso· 0 citations
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