In this paper we prove some regularity properties of solutions to variational inequalities of the form ${\int }_{\Omega }〈\mathcal{A}(x,u,Du),D(\phi -u)〉dx\ge {\int }_{\Omega }\mathcal{B}(x,u,Du)(\phi -u)dx,\forall \phi \in {\mathcal{K}}_{\psi }\left(\Omega \right).$Here $\Omega$ is a bounded open set of ${\mathbb{R}}^n$, $n\ge 2$, the function $\psi :\Omega \to [-\infty ,+\infty )$, called obstacle, belongs to the Sobolev class $W^{1,p}\left(\Omega \right)$ and ${\mathcal{K}}_{\psi }\left(\Omega \right)=\{w\in W^{1,p}\left(\Omega \right):w\ge \psi \text{q.o. in}\Omega \}$ 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 $L^{pq}$, for every $q\in (1,\infty )$, provided the partial map $(x,u)\mapsto \mathcal{A}(x,u,\xi )$ is Hölder continuous and $\mathcal{B}(x,u,\xi )$ 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.

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 ${\int }_{\Omega }〈\mathcal{一种}(X,你,D你),D(\phi -你)〉dX\ge {\int }_{\Omega }\mathcal{乙}(X,你,D你)(\phi -你)dX,\forall \phi \in {\mathcal{钾}}_{\psi }\left(\Omega \right).$这里 $\Omega$ 是一个有界开集 ${\mathbb{电阻}}^n$, $n\ge 2$， 功能 $\psi ：\Omega \to [-\infty ,+\infty )$，称为障碍物，属于 Sobolev 类${宽}^{1,磷}\left(\Omega \right)$ 和 ${\mathcal{钾}}_{\psi }\left(\Omega \right)=\{瓦\in {宽}^{1,磷}\left(\Omega \right)：瓦\ge \psi \text{输入}\Omega \}$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${升}^{磷q}$，对于每 $q\in (1,\infty )$, 提供部分地图 $(X,你)\mapsto \mathcal{一种}(X,你,\xi )$ 是 Hölder 连续的并且 $\mathcal{乙}(X,你,\xi )$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。