71 research outputs found

    Oblique boundary value problems for augmented Hessian equations I

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    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

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    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

    New maximum principles for linear elliptic equations

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    We prove extensions of the estimates of Aleksandrov and Bakel′'man for linear elliptic operators in Euclidean space Rn\Bbb{R}^{\it n} to inhomogeneous terms in LqL^q spaces for q<nq < n. 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 L2L^2 estimates

    Hessian measures II

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    In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain \Om in Euclidean n-space, k=1,...,n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.Comment: 26 pages, published versio
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