Weighted Global Regularity Estimates for Elliptic Problems with Robin Boundary Conditions in Lipschitz Domains
报告题目(Title)：Weighted Global Regularity Estimates for Elliptic Problems with Robin Boundary Conditions in Lipschitz Domains
报告人(Speaker)：杨四辈 教授 （兰州大学）
地点(Place)：腾讯会议 726 539 520
时间(Time)：2020年6月12日 15:00-16:00
邀请人(Inviter)：杨大春
报告摘要
Let $n\ge2$ and $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this talk, we introduce 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 $\Omega$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, we 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 H\"older 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, we further 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. By this and some technique from harmonic analysis, we further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak--)Orlicz spaces and variable Lebesgue spaces. This talk is based on a joint work with Profs. Dachun Yang and Wen Yuan.
