论文标题
Lipschitz域中罗宾边界条件的椭圆形问题的加权全球规律性估计
Weighted Global Regularity Estimates for Elliptic Problems with Robin Boundary Conditions in Lipschitz Domains
论文作者
论文摘要
令$ n \ ge2 $和$ω$为$ \ mathbb {r}^n $中的一个有界的Lipschitz域。在本文中，作者研究了解决方案梯度的全局（加权）估计值，以换句形式的二阶椭圆形方程，具有实现的，有限的，可测量的系数为$ω$。更确切地说，让$ p \ in（n/（n/（n-1），\ infty）$。使用真实的论点，作者分别获得了$ w^{1，p} $的两个必要条件，分别估计了罗宾边界价值问题的解决方案，以较弱的反向hölder不平等为指数$ p $ $ p $或加权$ w^{1，q} $估算的解决方案的估计值$ q \ in（n/（n/（n/（n/（n/（n/（N/））），作者为解决方案的解决方案建立了一些全球规则性估计，以分别在有限的Lipschitz域上，分别在$ c^1 $ tomains或（半）凸面域上，在有限的lebes sace上，分别在有限的范围内，分别在有限的lipschitz域上，分别在有限的lipschitz域上，分别在有限的lebes sace上，这是一个很大程度上的方法，这些方法是在某些方面的规模，这是一个很大的方法，这是一个很大程度上，这是某些方法，这是某些方法，这是一个很大程度上的方法$ n = 3 $在有界的$ c^1 $域中，还为此提供了某些知识的替代证明。
Let $n\ge2$ and $Ω$ be a bounded Lipschitz domain in $\mathbb{R}^n$. 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 in $Ω$. More precisely, let $p\in(n/(n-1),\infty)$. 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 $q\in(n/(n-1),p]$ 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 $\mathrm{BMO}$ 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 know result. 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
