关于低维Sobolev同胚的两个问题 献给余家荣教授100华诞

On two questions of Sobolev homeomorphisms in low dimensions

本文讨论和介绍Sobolev同胚的一些性质及相关结果,证明一维情形下微分同胚在Sobolev同胚中的稠密性,还证明在固定体积形式和边界的约束下,如果二维圆盘上新的体积形式是径向对称的并且一致靠近于Lebesgue测度,那么旋转对称同胚是Dirichlet能量的唯一极小解.

We present some properties of Sobolev homeomorphisms. We prove the density of diffeomorphisms in the space of Sobolev homeomorphisms in one-dimensional case. On the unit disc of R^2, under the constraints of the prescribed volume form and fixed boundary, we show that if the volume form is radially symmetric and uniformly close to the Lebesgue measure, then the only radial homeomorphism is the unique minimizer for the Dirichlet energy.