Gradient estimates for divergence form parabolic systems

2020/05/17 by Hongjie Dong, Dong, Hongjie, Longjuan Xu +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2005.08157

openalex publication_date 2020/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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,Dini and C^γ₀ 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 \citell,fknn to a large extent. We also prove a global weak type-(1,1) estimate with respect to A₁ 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,Dini interfaces. This gives an extension of a recent result in \citecss to parabolic systems.

Citations

• On the C 1 regularity of solutions to divergence form elliptic systems with dini-continuous coefficients

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