83 research outputs found
Oblique boundary value problems for augmented Hessian equations I
In this paper, we study global regularity for oblique boundary value problems
of augmented Hessian equations for a class of general operators. By assuming a
natural convexity condition of the domain together with appropriate convexity
conditions on the matrix function in the augmented Hessian, we develop a global
theory for classical elliptic solutions by establishing global a priori
derivative estimates up to second order. Besides the known applications for
Monge-Amp`ere type operators in optimal transportation and geometric optics,
the general theory here embraces prescribed mean curvature problems in
conformal geometry as well as oblique boundary value problems for augmented
k-Hessian, Hessian quotient equations and certain degenerate equations.Comment: Revised version containing minor clarification
On the second boundary value problem for Monge-Ampere type equations and geometric optics
In this paper, we prove the existence of classical solutions to second
boundary value prob- lems for generated prescribed Jacobian equations, as
recently developed by the second author, thereby obtaining extensions of
classical solvability of optimal transportation problems to problems arising in
near field geometric optics. Our results depend in particular on a priori
second derivative estimates recently established by the authors under weak
co-dimension one convexity hypotheses on the associated matrix functions with
respect to the gradient variables, (A3w). We also avoid domain deformations by
using the convexity theory of generating functions to construct unique initial
solutions for our homotopy family, thereby enabling application of the degree
theory for nonlinear oblique boundary value problems.Comment: Final version to appear in Archive for Rational Mechanics and
Analysi
On the Dirichlet problem for general augmented Hessian equations
In this paper we apply various first and second derivative estimates and
barrier constructions from our treatment of oblique boundary value problems for
augmented Hessian equations, to the case of Dirichlet boundary conditions. As a
result we extend our previous results on the Monge-Ampere and k-Hessian cases
to general classes of augmented Hessian equations in Euclidean spaceComment: This paper is a spin-off from our treatment of oblique boundary value
problems which was first posted on arXiv in 2015 and replaces earlier draft
New maximum principles for linear elliptic equations
We prove extensions of the estimates of Aleksandrov and Bakelman for
linear elliptic operators in Euclidean space to inhomogeneous
terms in spaces for . Our estimates depend on restrictions on the
ellipticity of the operators determined by certain subcones of the positive
cone. We also consider some applications to local pointwise and
estimates
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