$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients

Abstract

We prove a global estimate in the Sobolev spaces with variable exponents to the solution of a class of higher-order divergence parabolic equations with measurable coefficients over the non-smooth domains. Here, it is mainly assumed that the coefficients are allowed to be merely measurable in one of the spatial variables and have a small BMO quasi-norm in the other variables at a sufficiently small scale, while the boundary of the underlying domain belongs to the so-called Reifenberg flatness. This is a natural outgrowth of Dong-Kim-Zhang’s papers [1, 2] from the $W^{m,p}$-regularity to the $W^{m,p(t,x)}$-regularity for such higher-order parabolic equations with merely measurable coefficients with Reifenberg flat domain which is beyond the Lipschitz domain with small Lipschitz constant.