论文标题
BMO抗对称零件的椭圆运算符的Dirichlet问题的全球梯度估计值
Global Gradient Estimates for Dirichlet Problems of Elliptic Operators with a BMO Anti-Symmetric Part
论文作者
论文摘要
令$ n \ ge2 $和$ω\ subset \ mathbb {r}^n $为有界的NTA域。在本文中，作者研究了（加权）全局梯度估计，以椭圆形零件和$ω$中的椭圆形对称零件和BMO反对称零件的二阶椭圆形方程的Dirichlet边界值问题。 More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems.作为应用，作者为解决方案的解决方案建立了一些全球梯度估算，以差异形式的二阶椭圆形方程，小$ \ m m i \ mathrm {bmo} $对称零件和小$ \ mathrm {bmo} $ andift inlipschitz domains，quasiains domains quasiains domains quasiains nofe ins quasiains nofears，加权lebesgue空间中的$ c^1 $域或（半）凸域。此外，随着进一步的应用，作者分别获得了（加权）洛伦兹空间（Lorentz-）Morrey空间，（Musielak-）Orlicz空间和可变的Lebesgue空间。即使在Lebesgue空间中的全球梯度估计中，本文获得的结果也通过削弱了对系数矩阵的假设来改善已知结果。
Let $n\ge2$ and $Ω\subset\mathbb{R}^n$ be a bounded NTA domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part in $Ω$. More precisely, for any given $p\in(2,\infty)$, the authors prove that a weak reverse Hölder inequality with exponent $p$ implies the global $W^{1,p}$ estimate and the global weighted $W^{1,q}$ estimate, with $q\in[2,p]$ and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ symmetric part and small $\mathrm{BMO}$ anti-symmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, $C^1$ domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz--)Morrey spaces, (Musielak--)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.
