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Regularity results for solutions to a class of obstacle problems
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 ΩA(x,u,Du),D(φu)dxΩB(x,u,Du)(φu)dx,φKψ(Ω).Here Ω is a bounded open set of Rn, n2, the function ψ:Ω[,+), called obstacle, belongs to the Sobolev class W1,p(Ω) and Kψ(Ω)={wW1,p(Ω):wψq.o. inΩ} 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 Lpq, for every q(1,), provided the partial map (x,u)A(x,u,ξ) is Hölder continuous and B(x,u,ξ) 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 Ω一种(X,,D),D(φ-)dXΩ(X,,D)(φ-)dX,φψ(Ω).这里 Ω 是一个有界开集 电阻n, n2, 功能 ψΩ[-,+),称为障碍物,属于 Sobolev 类1,(Ω)ψ(Ω)={1,(Ω)ψ输入Ω}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q,对于每 q(1,), 提供部分地图 (X,)一种(X,,ξ) 是 Hölder 连续的并且 (X,,ξ)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。

更新日期:2021-06-14
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