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 $$ D^2u\in L_{\mathrm{loc}}^{P,\infty}, \qquad P=\frac{1+\lambda}{1-\lambda}. $$ This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent $P$. We further construct strictly convex $C^1$ solutions for which the distribution function of the Laplacian is comparable to $T^{-P}$ for all sufficiently large $T$. Hence both the exponent and the weak Lorentz endpoint are optimal: $L^{P,\infty}$ cannot be replaced by $L^{P,b}$ for any finite $b>0$. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.
We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\emph{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $\kappa=\Lambda/\lambda$, then the optimal exponent in the $W^{2,\epsilon}$ regul...
Thialita M. Nascimento, Eduardo V. Teixeira· 0 citations
We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enla...
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 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...
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...
In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$\sigma_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overline{\Omega}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\Omega\Subset\R^n$, we establish ball rigidity results in al...
Zhi-Hui Zhang· 0 citations
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