Global gradient estimates for parabolic systems from composite materials
The authors prove regularity estimates for weak solutions of parabolic systems with measurable coefficients in domains that are finite unions of subdomains with Reifenberg flat boundaries, precisely \(\Omega\) contains a finite union of subdomains \(\Omega_l, l = 1, \ldots, K\), whose boundaries are Reifenberg flat and \[ u_t^i - D_{\alpha}[A_{ij}^{\alpha \beta}D_{\beta} u^j ]= D_{\alpha}F^i_{\alpha}, \text{ in } \Omega_T = \Omega \times (0, T], \] \[ u^i = 0, \text{ on } \partial_p \Omega_T = \partial \Omega \times [0,t] \cup \Omega \times \{t = 0 \}. \] The coefficients are assumed to have BMO norm that is small in each subdomain, but can have large jumps on the boundaries. They remark that the estimates do not depend on the distance between the subdomains, and that the subdomains are allowed to touch.The results are proved for Orlicz norms that include results for \(L^p\). For example, if \(p \geq 2\), there is a \(\delta\), which satisfies \(0 < \delta < 1/8\), such that if the coefficients are \((\delta, R) \) composite vanishing and \(F\) is a vector field in \(L^p\), then \[ \int_0^T \int_{\Omega} |Du|^p \, dx dt \leq C \int_0^T \int_{\Omega} |F|^p \, dx dt. \] Here \((\delta, R)\) composite vanishing means that the BMO norm of the coefficients (now denoted \(A_{ij}^{\alpha \beta, l}\) on \(\Omega_l)\) is less than or equal to \(\delta\) over balls of radius less than \(R\) in each subdomain \(\Omega_l\). The BMO condition applies to each subdomain, but the BMO norm could be large if considered on \(\Omega\).
scientific article; zbMATH DE number 6873459
scientific article