论文标题
在梯度上具有给定边界数据和凸约限制的功能
On functions with given boundary data and convex constraints on the gradient
论文作者
论文摘要
令$ω\ subset \ mathbb {r}^{d} $为打开集。给定$ \partialΩ$上的边界数据$ g $和一个函数$ k：\ barω\ to \ mathcal {k} $，所有紧凑型凸的家族的$ \ mathbb {r}^{d {d} $的家族，我们证明了函数的存在$：ω\ to \ tho \ to \ to $ u： $ \ nabla u（x）\ in K（x）$ a.e。我们研究了集合$ \ MATHCAL {u} \ subset \barΩ$的定期，它们都重合。
Let $Ω\subset\mathbb{R}^{d}$ be an open set. Given a boundary datum $g$ on $\partialΩ$ and a function $K:\bar Ω \to\mathcal{K}$, the family of all compact convex subsets of $\mathbb{R}^{d}$, we prove the existence of functions $u:Ω\to\mathbb{R}$ such that $u=g$ on $\partialΩ$ and $\nabla u(x)\in K(x)$ a.e. and we investigate the regularity of such solutions on the set $\mathcal{U} \subset \barΩ$ of points at which they all coincide.
