부산대학교 도서관

Abstraact. ``This article has two purposes. The first is to prove thatsolutions of the second boundary value problem for generated Jacobianequations (GJEs) are strictly $g$-convex. The second is to prove theglobal $C^3$ regularity of Aleksandrov solutions to the same problem.In particular, Aleksandrov solutions are classical solutions. These arerelated because the strict $g$-convexity is essential for the proof ofthe global regularity. The assumptions for the strict $g$-convexity arethe natural extension of those used by Chen and Wang in the optimaltransport case. They are the Loeper maximum principle condition, apositively pinched right-hand side, a $g^*$-convex target, and a sourcedomain strictly contained in a $g$-convex domain. This improves theexisting domain conditions, though at the expense of requiring a $C^3$generating function. This is appropriate for global regularity whereexistence is proved assuming a $C^4$ generating function. We prove theglobal regularity under the hypothesis that Jiang and Trudingerrecently used to obtain the existence of a globally smooth solution andan additional condition on the height of solutions. Our proof of globalregularity is by modifying Jiang and Trudinger's existence result toconstruct a globally $C^3$ solution intersecting the Aleksandrovsolution. Then the strict convexity yields the interior regularity toapply the author's uniqueness results.''