On Second Order Elliptic and Parabolic Equations of Mixed Type

Gong Chen; Mikhail Safonov

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arXiv · 1401.0351

On Second Order Elliptic and Parabolic Equations of Mixed Type

Gong Chen·Mikhail Safonov

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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.

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Gong Chen, Mikhail Safonov. 2014-01-02. On Second Order Elliptic and Parabolic Equations of Mixed Type. https://arxiv.org/abs/1401.0351

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