Some Multi-valued Solutions to Monge–Ampère Equations

Entire Invariant Solutions to Monge-ampère Equations

We prove existence and regularity of entire solutions to MongeAmpère equations invariant under an irreducible action of a compact Lie group. We consider Monge-Ampère equations of the form f(∇φ) detDijφ = g(x) (0.1) where f and g are nonnegative measurable functions on R. We recall first the concept of a weak solution of (0.1). Let φ be a convex function. Then ∇φ is a well-defined multi-valued m...

Boundary Regularity for Solutions to the Linearized Monge-ampère Equations

We obtain boundary Hölder gradient estimates and regularity for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are affine invariant analogues of the boundary Hölder gradient estimates of Krylov.

Existence of convex solutions for systems of Monge-Ampère equations

Convex Solutions of Systems Arising from Monge-ampère Equations

We establish two criteria for the existence of convex solutions to a boundary value problem for weakly coupled systems arising from the Monge-Ampère equations. We shall use fixed point theorems in a cone.

Global existence for the semigeostrophic equations via Sobolev estimates for Monge-Ampère

These notes record and extend the lectures for the CIME Summer Course held by the author in Cetraro during the week of June 2-7, 2014. Our goal is to show how some recent developments in the theory of the Monge-Ampère equation play a crucial role in proving existence of global weak solutions to the semigeostrophic equations.