Alexandrov’s estimate states that if $\Omega $ is a bounded open convex domain in $\mathbb {R}^n$ and $u:\bar \Omega \to \mathbb {R}$ is a convex solution of the Monge-Ampère equation $\det D^2 u = f$ that vanishes on $\partial \Omega $, then

We establish a variety of improvements of this, depending on the geometry of $\partial \Omega $. For example, we show that if the curvature is bounded away from $0$, then the estimate remains valid if $\omega (\delta )$ is replaced by $C_\Omega \delta ^{\frac 12 + \frac 1{2n}}$. We determine the sharp constant $C_\Omega $ when $n=2$, and when $n\ge 3$ and $\partial \Omega $ is $C^2$, we determine the sharp asymptotics of the optimal modulus of continuity $\omega _\Omega (\delta )$ as $\delta \to 0$. For arbitrary convex domains, we characterize the scaling of the optimal modulus $\omega _\Omega $. Our results imply in particular that unless $\partial \Omega $ has a flat spot, $\omega _\Omega (\delta ) = o(\delta ^{1/n})$ as $\delta \to 0$, and under very mild nondegeneracy conditions, they yield the improved Hölder estimate, $\omega _\Omega (\delta ) \le C \delta ^\alpha $ for some $\alpha>1/n$.

The space of Monge–Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\,\mapsto \,\text{Det}\,{{\text{D}}^{2}}u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge–Ampère functions. We also prove that if a Monge–Ampère function $u$ on a bounded set $\Omega \,\subset \,{{\mathbb{R}}^{2}}$ satisfies the equation $\text{Det}\,{{D}^{2}}u\,=\,0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge–Ampère function of 2 variables.

This paper gives a rigorous derivationof a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filamentsin a rotationally forced Bose-Einstein condensate. Thisfunctional is derived as a Γ-limitof scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence of large numbers of nontrivial local minimizers and we prove that, given any suchlocal minimizer, the Gross-Pitaevsky functionalhas a local minimizer that is nearby (in a suitable sense) whenever a scalingparameter is sufficiently small.