$W^{m, p(t, x)}$-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.

Keywords

A higher-order parabolic equation / Sobolev spaces with variable exponents / partially BMO quasi-norm / Reifenberg flat domains / log-Hölder continuity