We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge-Ampere method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis f...

This is the first paper in a series to develop linear and nonlinear theory for elliptic parabolic equations on Kahler varieties with mild singularities. Donaldson has established Schauder estimate complex Monge-Ampere when background metrics $\mathbb{C}^n$ have cone singularities along smooth hypersurface. We prove sharp pointwise metric $g_\beta= \sqrt{-1} ( dz_1 \wedge d\bar{z_1} + \ldots \be...

Let H be Monge-Ampère singular integral operator, [Formula: see text], and [Formula: see text]. It is proved that the commutator [Formula: see text] is bounded from [Formula: see text] to [Formula: see text] for [Formula: see text] and from [Formula: see text] to [Formula: see text] for [Formula: see text]. For the extreme case [Formula: see text], a weak estimate is given.

We solve the Dirichlet problem for complex Monge–Ampère equation near an isolate Klt singularity, which generalizes result of Eyssidieux et al. [Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607–639], where is solved on singular varieties without boundary. As a corollary, we construct solutions to with isolated singularity strongly pseudoconvex domain [Formula: see text] conta...

A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation. The method improves several inequalities literature, e.g. constants a theorem of Cabre--Ros--Oton--Serra. Applications are given context generalization log-convex density conjecture due to Brakke and resolved by Chambers: case $\alpha-$homogeneous ($\alpha>0$), concave densities, (mod transla...

are smooth, given that k is smooth and nonnegative. When u is radial, (1) reduces to a nonlinear ODE on [0, 1) that is singular at the endpoint 0. It is thus easy to prove that u is always smooth away from the origin, even where k vanishes, but smoothness at the origin is more complicated, and determined by the order of vanishing of k there. In fact, Monn [9] proves that if k = k (x) is indepen...

On a bounded strictly pseudoconvex domain in $\Bbb{C}^n$, $n>1$, the smoothness of Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up boundary is obstructed by local curvature invariant boundary. For domains $\Bbb{C}^2$ which are diffeomorphic ball, we motivate and consider problem determining whether global vanishing this obstruction implies biholomorphic equivalence unit ball....