John Urbas

The Dirichlet problem for the equation of prescribed Gauss curvature

We treat necessary and sufficient conditions for the classical solvability of the Dirichlet problem for the equation of prescribed Gauss curvature in uniformly convex domains in Euclidean n space. Our methods simultaneously embrace more general equations of Monge-Ampere type and we establish conditions which ensure that solutions have globally bounded second derivatives.

Regularity of generalized solutions of Monge-Ampère equations

On etudie la regularite interieure des solutions generalisees convexes des equations de Monge Ampere de la forme detD 2 d=f(x,u,Du) dans Ω, ou Ω est un domaine de R n , f est une fonction positive suffisamment lisse sur Ω×R×R n

Interior curvature bounds for a class of curvature equations

We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Amp` ere type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C 1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

On second derivative estimates for equations of Monge-Ampère type

We derive interior second derivative estimates for solutions of equations of Monge-Ampere type which extend those of Pogorelov for the case of affine boundary values. A key ingredient in our method is the existence of a strong solution of the homogeneous Monge-Ampere equation.

Nonlinear oblique boundary value problems for Hessian equations in two dimensions

Abstract We study nonlinear oblique boundary value problems for nonuniformly elliptic Hessian equations in two dimensions. These are equations whose principal part is given by a suitable symmetric function of the eigenvalues of the Hessian matrix D 2 u of the solution u . An interesting feature of our second derivative estimates is the need for certain strong structural hypotheses on the boundary condition, which are not needed in the uniformly elliptic case. Restrictions of this type are natural in our context; we present examples showing that second derivative bounds may fail if we do not assume such conditions.

An interior second derivative bound for solutions of Hessian equations

Abstract. In previous work we showed that weak solutions in

The secondary boundary value problem for a class of Hessian equations

W^{2,p}(\Omega)

Global Hölder estimates for equations of Monge-Ampère type

of the k-Hessian equation

The generalized Dirichlet problem for equations of Monge-Ampère type

F_k[u]=g(x)

Some new estimates for solutions of degenerate two dimensional Monge-Ampère equations

have locally bounded second derivatives if g is positive and sufficiently smooth and p > kn/2. Here we improve this result to p > k(n-1)/2, which is known to be sharp in the Monge-Ampère case k=n > 2.