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

Oblique boundary value problems for augmented Hessian equations II

In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented Hessian is only assumed regular. This enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case. Finally we give some applications to existence theorems which embrace standard Hessian equations as special cases.Research supported by National Natural Science Foundation of China (No. 11401306), Australian Research Council (No. DP1094303), China Postdoctoral Science Foundation (No. 2015M571010) and Jiangsu Natural Science Foundation of China (No. BK20140126)

It Is Not Just What We Say, But How We Say Them: LDA-based Behavior-Topic Model

Textual information exchanged among users on online social network platforms provides deep understanding into user-s ’ interest and behavioral patterns. However, unlike tradi-tional text-dominant settings such as offline publishing, one distinct feature for online social network is users ’ rich inter-actions with the textual content, which, unfortunately, has not yet been well incorporated in the existing topic modeling frameworks. In this paper, we propose an LDA-based behavior-topic mod-el (B-LDA) which jointly models user topic interests and be-havioral patterns. We focus the study of the model on online social network settings such as microblogs like Twitter where the textual content is relatively short but user interactions on them are rich. We conduct experiments on real Twitter data to demonstrate that the topics obtained by our model are both informative and insightful. As an application of our B-LDA model, we also propose a Twitter followee rec-ommendation algorithm combining B-LDA and LDA, which we show in a quantitative experiment outperforms LDA with a significant margin.