On second order elliptic and parabolic equations of mixed type

It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are Holder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case (x is an element of R-1), these properties are not preserved for equations of mixed divergence nondivergence structure: for elliptic equations. D-i(a(ij)(1)D(ju))+a(ij)(2)D(iju) =O, and parabolic equations p partial derivative(tu) = Di(a(ij)D(ju)), where p = p(t,x) is a bounded strictly positive function. The Hi51der continuity and Harnack inequality are known if p does not depend either on t or on x. We essentially use homogenization techniques in our construction. Bibliography: 22 titles. (C)2017 Elsevier Inc. All rights reserved.