Regularity results for a class of obstacle problems with p, q−growth conditions
Regularity results for a class of obstacle problems with p, q−growth conditions
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具有 p、q−增长条件的一类障碍问题的规律性结果
DOI：
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发表时间：
2019
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通讯作者：
A. Napoli
中科院分区：
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作者：
Michele Caselli;M. Eleuteri;A. Napoli
在本文中，我们证明了一类障碍问题的解的局部 Lipschitz 连续性
min{ ∫ΩF(x, Dz) : z ∈ ?ψ(Ω)}。
这里 ?ψ(Ω) 是对于给定的 u0 ∈ W1,p(Ω) 的允许函数 z ∈ u0 + W1,p(Ω) 的集合，使得 z ≥ ψ a.e.在 Ω 中，ψ 是障碍物，Ω 是 ℝn, n ≥ 2 的开有界集。这里的主要新颖之处在于我们假设被积函数 F(x, Dz) 满足 (p, q) 增长条件，并且作为 x 变量的函数属于合适的 Sobolev 类。我们注意到 Lipschitz 连续性结果是在增长指数和椭圆率指数之间非常接近的条件下获得的。此外，相对于之前的规律性结果，我们对障碍物施加了较少的限制性假设。此外，假设障碍ψ是局部有界的，我们证明了一大类主要部分满足非标准增长条件的变分不等式解的局部有界性。
In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type
min{ ∫ΩF(x, Dz) : z ∈ ?ψ(Ω)}.
Here ?ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.