Nonlinear Analysis: Real World Applications ( IF 1.8 ) Pub Date : 2021-06-14 , DOI: 10.1016/j.nonrwa.2021.103377 Antonio Giuseppe Grimaldi
In this paper we prove some regularity properties of solutions to variational inequalities of the form Here is a bounded open set of , , the function , called obstacle, belongs to the Sobolev class and is the class of the admissible functions. First we establish a local Calderòn–Zygmund type estimate proving that the gradient of the solutions is as integrable as the gradient of the obstacle in the scale of Lebesgue spaces , for every , provided the partial map is Hölder continuous and satisfies a suitable growth condition. Next, this estimate allows us to prove that a higher differentiability in the scale of Besov spaces of the gradient of the obstacle transfers to the gradient of the solutions.
中文翻译:
一类障碍问题解的正则性结果
在本文中,我们证明了以下形式的变分不等式的解的一些正则性质 这里 是一个有界开集 , , 功能 ,称为障碍物,属于 Sobolev 类 和 是可容许函数的类。首先,我们建立局部 Calderòn-Zygmund 类型估计,证明解的梯度与 Lebesgue 空间尺度中障碍物的梯度一样可积,对于每 , 提供部分地图 是 Hölder 连续的并且 满足适宜的生长条件。接下来,这个估计使我们能够证明障碍物梯度的 Besov 空间尺度的更高可微性转移到解的梯度。