On $$C^1$$ C 1 , $$C^2$$ C 2 , and weak type-(1, 1) estimates for linear elliptic operators: part II

On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients

“…Elliptic and parabolic equations with Dini mean oscillation coefficients considered in this paper were recently studied in [11,7,6,8]. Using Hölder's inequality, it is easily seen that the Dini mean oscillation condition (or L 1 -Dini mean oscillation condition) considered here can be derived from the L p -Dini mean oscillation condition for any p > 1, i.e., the function…”

“…The DMO condition considered here is one of the weakest conditions to guarantee the continuous differentiability of solutions. As in [11,7,6,8], the proof of Theorem 1.1 is based on Campanato s approach, which was used previously, for instance, in [14,18]. The main step of Campanato s approach is to show the mean oscillation of Du in balls (or cylinders) vanishes to a certain order as the radii of the balls (or cylinders) go to zero.…”

On the regularity of very weak solutions for linear elliptic equations in divergence form

“…Dini-continuous coefficients For the proof of our result, we will need the following extension of the Schauder regularity theory for elliptic equations in divergence form with Dini continuous coefficients (see [12,Theorem 1.1] and [6,Theorem 1.3], see also [11] which is inclusive of the parabolic case.). For the L p -regularity theory we refer to [7], where the general case of V MO coefficients is treated (see also […”

“…The assumption in [6] about the coefficients is weaker then (1.4), since they assume that the modulus of continuitỹ…”

Gradient estimates for Stokes systems with Dini mean oscillation coefficients

“…Let us briefly describe our arguments based on Campanato's approach. Such approach was used in [16,21] and further developed in [8,10,6,14]. The key point is to show the mean oscillations of Du and p in balls vanish in a certain order as the radii of balls go to zero.…”