Oldřich John

A note on the regularity of autonomous quasilinear elliptic and parabolic systems

An analogy of E.Giusti lemma at infinity was introduced in [5]. Its slight modification enables us to prove the estimates of Holder norms for BMO weak solutions of autonomous elliptic and parabolic systems provided all such solutions are continuous at the same point. The sharpness of this condition is illustrated by a counterexample

Interior C 1, γ-Regularity for Weak Solutions of Nonlinear Second Order Parabolic Systems

The interior C γ, γ/2-regularity for the first gradient with respect to the space variables of a weak solution to a class of nonlinear second order parabolic systems is proven under the assumption that oscillations of coefficients are controlled by the ellipticity constant.