论文标题
发散形式抛物线系统的梯度估计值
Gradient estimates for divergence form parabolic systems
论文作者
论文摘要
我们考虑在一个圆柱域中的二阶抛物面系统，具有有限数量的子域的二阶形式，假设界面边界为$ c^{1，\ text {dini}} $和$ c^{γ_{0}} $在空间变量和时间变量中，相应的。梯度估计和分段$ c^{1/2,1} $ - 假定领先的系数和数据为分段二型excillation或分段Hölder连续时，建立了规律性。我们的结果在很大程度上改善了\ cite {ll，fknn}的结果。对于$ a_ {1} $ muckenhoupt的重量，我们还证明了全球弱类型-A_ {1} $（1,1）$估计，用于抛物线系统的抛物线系统具有领先的系数，这些系数满足了更强的假设。作为副产品，我们提供了$ c^{1，μ} $或$ c^{1，\ text {dini}} $ Interfaces的最佳定期弱解决方案。这给出了\ cite {css}的最新结果扩展到抛物线系统。
We consider divergence form, second-order strongly parabolic systems in a cylindrical domain with a finite number of subdomains under the assumption that the interfacial boundaries are $C^{1,\text{Dini}}$ and $C^{γ_{0}}$ in the spatial variables and the time variable, respectively. Gradient estimates and piecewise $C^{1/2,1}$-regularity are established when the leading coefficients and data are assumed to be of piecewise Dini mean oscillation or piecewise Hölder continuous. Our results improve the previous results in \cite{ll,fknn} to a large extent. We also prove a global weak type-$(1,1)$ estimate with respect to $A_{1}$ Muckenhoupt weights for the parabolic systems with leading coefficients which satisfy a stronger assumption. As a byproduct, we give a proof of optimal regularity of weak solutions to parabolic transmission problems with $C^{1,μ}$ or $C^{1,\text{Dini}}$ interfaces. This gives an extension of a recent result in \cite{css} to parabolic systems.
