Estimates for the highest derivatives of solutions of elliptic and parabolic equations with continuous coefficients

We show that, in a certain sense, the Dini condition on the modulus of continuity of the coefficients of elliptic and parabolic equations of arbitrary order in a domain is necessary both for the existence of a classical solution of the first boundary problem under arbitrarily smooth data, and for deriving an a priori estimate for the maximum of the moduli of the highest derivatives in a do man ' in terms of max¦u¦in . The bibliography contains seven references.Translated from Matematicheski Zametki, Vol. 2, No. 5, pp. 549–560, November, 1967.