부산대학교 도서관

Abstraact. ``This article has two purposes. The first is to prove that solutions of the second boundary value problem for generated Jacobian equations (GJEs) are strictly $g$-convex. The second is to prove the global $C^3$ regularity of Aleksandrov solutions to the same problem. In particular, Aleksandrov solutions are classical solutions. These are related because the strict $g$-convexity is essential for the proof of the global regularity. The assumptions for the strict $g$-convexity are the natural extension of those used by Chen and Wang in the optimal transport case. They are the Loeper maximum principle condition, a positively pinched right-hand side, a $g^*$-convex target, and a source domain strictly contained in a $g$-convex domain. This improves the existing domain conditions, though at the expense of requiring a $C^3$ generating function. This is appropriate for global regularity where existence is proved assuming a $C^4$ generating function. We prove the global regularity under the hypothesis that Jiang and Trudinger recently used to obtain the existence of a globally smooth solution and an additional condition on the height of solutions. Our proof of global regularity is by modifying Jiang and Trudinger's existence result to construct a globally $C^3$ solution intersecting the Aleksandrov solution. Then the strict convexity yields the interior regularity to apply the author's uniqueness results.''