Related papers: Quantitative stability for the complex Monge-Amp\`…
The existence and multiplicity and nonexistence of nontrivial radial convex solutions of systems of Monge-Amp\`ere equations are established with superlinearity or sublinearity assumptions for an appropriately chosen parameter. The proof of…
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
We survey the (old and new) regularity theory for the Monge-Amp\`ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\`ere type equations arising in that…
Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Amp\`ere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are H\"older continuous with…
Existence and boundary regularity away from the corners are established for two-dimensional Monge-Amp\`{e}re equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions…
We investigate different measures of stability of quantum statistical ensembles with respect to local measurements. We call a quantum statistical ensemble "stable" if a small number of local measurements cannot significantly modify the…
In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Amp\`ere operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Amp\`ere…
This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory,…
The consistency of the concept of quantum (quasi)particles possessing effective mass which is both position- and excitation-dependent is analyzed via simplified models. It is shown that the system may be stable even when the effective mass…
We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Amp\`ere equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating…
In this paper, we establish boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,\gamma}$ boundary values under natural assumptions on the…
We prove stability of solutions of the complex Monge-Amp\`ere equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be H\"older…
The Monge-Amp\`{e}re equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the…
We study continuity properties of generalized Monge-Amp\`ere operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a…
Given a compact K\"ahler manifold, we survey the study of complex Monge-Amp\`ere type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of…
We study the parabolic complex Monge-Amp\`ere type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a…
We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries small perturbations can lead to large deviations over long times, while…
We give a sufficient condition on a sequence of uniformly bounded $\omega$-plurisubharmonic functions, $\omega$ being a Hermitian metric, for which the sequence of associated Monge-Amp\`ere measures converges weakly. This criterion can be…
We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a…
We construct new examples of Monge-Amp\`{e}re metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular…