The obstacle problem for a fractional Monge–Ampère equation

Abstract

We study the obstacle problem for a nonlocal, degenerate elliptic Monge–Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.     

A complex parabolic type monge-ampère equation

The complex parabolic type Monge-Ampère equation we are dealing with is of the form inB × (0,T),u=ϕ on Γ, whereB is the unit ball in ℂ ^d ,d>1, and Γ is the parabolic boundary ofB × (0,T). Solutionu is proved unique in the class .     

Classical solvability of the Dirichlet problem for the Monge-Ampère equation

It is proved that the problemdet(u _xx)=f(x,u,u _x)>0, is solvable in spaces , provided a natural connection between the curvature of the closed surface and the growth of the functionf(x,u,p) in¦p¦ is valid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 72–79, 1983.It is my pleasure mentioning that I have discussed the above material many times with O. A. Ladyzhenskaya and that for the clear understanding of all aspects of the problem I am deeply indebted to her for her remarks and advice.     

Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.     

Some new estimates for the complex Monge-Ampère equation

In this paper, we consider the complex Monge-Ampère equation posed on a compact K?hler manifold. We show how to get L~p(p ∞) and L∞estimates for the gradient of the solution in terms of the continuity of the right-hand side.     

On the Dirichlet problem for Monge-Ampère type equations

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampére type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in Ann. Scoula Norm. Sup. Pisa Cl. Sci. VIII, 143–174 2009 in their study of global regularity in optimal transportation as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.     

Global solvability of monge-ampère type equations

We study the global solvability of Monge-Ampère equations of mixed type by "blowing up" the problem onto the torus embedded at the singular point of the equations     

Existence and nonexistence to exterior Dirichlet problem for Monge–Ampère equation

We consider the exterior Dirichlet problem for Monge–Ampère equation with prescribed asymptotic behavior. Based on earlier work by Caffarelli and the first named author, we complete the characterization of the existence and nonexistence of solutions in terms of their asymptotic behaviors.     

A meshfree method for solving the Monge–Ampère equation

Numerical Algorithms - This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and...     

Integrable structures for a generalized Monge-Ampère equation

We consider a third-order generalized Monge-Ampère equation u_yyy ? u _xxy ² + u_xxxu_xyy = 0, which is closely related to the associativity equation in two-dimensional topological field theory. We describe all integrable structures related to it: Hamiltonian, symplectic, and also recursion operators. We construct infinite hierarchies of symmetries and conservation laws.     

The problem of regular isometric embeddings and the Monge-Ampère equation of hyperbolic type

A relationship between classical approaches to embeddings and the Monge-Ampère equations is described. A new method of constructing smooth solutions of the general Monge-Ampère equation of hyperbolic type for the domains of finite-stripe type is presented. Bibliography: 18 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 177–186.     

W 2,1 regularity for solutions of the Monge–Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is $W^{2,1}_{\mathrm{loc}}$ . This is obtained by showing higher integrability a priori estimates for D ² u, namely D ² u∈Llog ^k L for any k∈?.     

On the exterior Dirichlet problem for the Monge–Ampère equation in dimension two

In this paper, we consider the Dirichlet problem for the Monge–Ampère equation on exterior domains in dimension two and prove a theorem on the existence of solutions with prescribed asymptotic behavior at infinity.     

An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace’s equation

Basing on the fundamental ideas of Babenko, we construct a fundamentally new, unsaturated, numerical method for solving the axially symmetric exterior Neumann problem for Laplace’s equation. The distinctive feature of this method is the absence of the principal error term enabling us to automatically adjust to every class of smoothness of solutions natural in the problem.     

Dirichlet problem for the Boussinesq–Love equation

We study the cases of unique solvability of the Dirichlet problem for the Boussinesq–Love equation.