$W^{m,p(t,x)}$-Estimate for a Class of Higher-Order Parabolic Equations with Partially BMO Coefficients (Journal of Partial Differential Equations, 2024)
We prove a global estimate in the Sobolev spaces with variable exponents tothe solution of a class of higher-order divergence parabolic equations with measurablecoefficients 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 havea small BMO quasi-norm in the other variables at a sufficiently small scale, while theboundary of the underlying domain belongs to the so-called Reifenberg flatness. Thisis a natural outgrowth of Dong-Kim-Zhang’s papers [1, 2] from the $W^{m,p}$-regularityto 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 domainwith small Lipschitz constant.