On higher regularity of Stokes systems with piecewise Hölder continuous coefficients

In this paper, we consider higher regularity of a weak solution ( u , p ) (u,p) to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise C s , δ Cs,δ in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in C s + 1 , μ Cs+1,μ , where s s is a positive integer, δ ∈ ( 0 , 1 ) δ ∈ (0,1) , and μ ∈ ( 0 , 1 ] μ ∈ (0,1] , we show that D u Du and p p are piecewise C s , δ μ C^{s,δ μ} , where δ μ = min { 1 2 , μ , δ } δ μ=min \{1/2,μ ,δ \} . Our result is new even in the 2D case with piecewise constant coefficients.