On Second Order Elliptic and Parabolic Equations of Mixed Type

TL;DR

This paper demonstrates that key regularity properties like Hölder continuity and Harnack inequality do not hold for certain mixed-type second order elliptic and parabolic equations, even in one dimension, using homogenization techniques.

Contribution

It reveals the failure of classical regularity properties for mixed divergence-nondivergence equations, extending understanding beyond previously known cases.

Findings

Hölder continuity fails for mixed equations in 1D

Harnack inequality does not hold for these mixed equations

Homogenization techniques are used to construct counterexamples

Abstract

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.