Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains
Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains
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Lipschitz 域中 Robin 边界条件椭圆问题的加权全局正则性估计
DOI：
10.1016/j.jde.2021.06.010
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发表时间：
2021
影响因子：
2.4
通讯作者：
Yuan Wen
中科院分区：
文献类型：
--
作者：
Yang Sibei;Yang Dachun;Yuan Wen
设n≥ 2，Ω是Rn的有界Lipschitz整环.本文研究了Ω上具有实值、有界、可测系数的二阶散度型椭圆型方程Robin边值问题解的梯度的整体（加权）估计.更精确地说，令p∈（n/（n− 1），∞）。利用实变量的方法，分别利用指数为p的弱逆Hölder不等式或q∈（n/（n− 1），p]和Muckenhoupt权的加权W1，q估计，得到了Robin边值问题解的W1，p估计的两个充要条件.作为应用，在加权Lebesgue空间的尺度下，通过一些与现有方法不同的精细方法，分别在有界Lipschitz域、C1域或（半）凸域上建立了具有小BMO系数的二阶散度型椭圆型方程Robin边值问题解的整体正则性估计，在有界C1整环的情况下，即使n= 3，在一个附加的假设下，也给出了某些已知结果的另一个正确的证明.利用这一方法和调和分析中的一些技巧，分别在Morrey空间、（Musielak-）Orlicz空间和变Lebesgue空间中得到了全局正则性估计.
Let n≥ 2 and Ω be a bounded Lipschitz domain of R n. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second-order elliptic equations of divergence form with real-valued, bounded, measurable coefficients on Ω. More precisely, let p∈(n/(n− 1),∞). Using a real-variable argument, the authors obtain two necessary and sufficient conditions for W 1, p estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1, q estimates of solutions with q∈(n/(n− 1), p] and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second-order elliptic equations of divergence form with small BMO coefficients, respectively, on bounded Lipschitz domains, C 1 domains or (semi-) convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when n= 3 in case of bounded C 1 domains, also gives an alternative correct proof of some known result under an additional assumption. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces,(Musielak–) Orlicz spaces, and variable Lebesgue spaces.