Cordes conditions and some alternatives for parabolic equations and discontinuous diffusion ∗

We consider parabolic equations in nondivergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. There are some conditions that ensure regularity of solutions of boundary value problems for second order equations and that are known as Cordes conditions (see Cordes (1956)). These conditions restricts the scattering of the eigenvalues of the matrix of the coefficients at higher derivatives. Related conditions from Talenti (1965), Koshelev (1982), Kalita (1989), Landis (1998), on the eigenvalues are also called Cordes type conditions. Gihman and Skorohod (1975) obtained a closed condition implicitly as a part of the proof of the uniqueness of a weak solution in Section 3 of Chapter 3. Cordes (1956) considered elliptic equations. Landis (1998) considered both elliptic and parabolic equations. Koshelev (1982) considered systems of elliptic equations of divirgent type and Hölder property of solutions. Kalita (1989) considered union of divergent and nondivirgent cases. Conditions from Cordes (1956) are such that they are not necessary satisfied even for constant non-degenerate matrices b, therefore, the condition for b = b(x) means that the corresponding inequalities are satisfied for all x0 for some non-degenerate matrix θ(x0) and