It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are H\"{ o}lder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case ($x\in R^1$), these properties are not preserved for equations of mixed divergence-nondivergence structure: for elliptic equations. \begin{ equation*} D_i(a^1_{ ij}D_ju)+a^2_{ ij}D_{ ij}u=0, \end{ equation*} and parabolic equations \begin{ equation*} p\partial_t u=D_i(a_{ ij}D_ju), \end{ equation*} where $p=p(t,x)$ is a bounded strictly positive function. The H\"{ o}lder 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: 23 titles.
Categories: math.AP