On second order elliptic and parabolic equations of mixed type

Gong Chen, Mikhail Safonov

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3 Scopus citations


It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are Hölder continuous and satisfy the interior Harnack inequality. We show that even in the one-dimensional case (x∈R1), these properties are not preserved for equations of mixed divergence–nondivergence structure: for elliptic equations. Di(aij1Dju)+aij2Diju=0, and parabolic equations p∂tu=Di(aijDju), where p=p(t,x) is a bounded strictly positive function. The Hölder 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.

Original languageEnglish (US)
Pages (from-to)3216-3237
Number of pages22
JournalJournal of Functional Analysis
Issue number8
StatePublished - Apr 15 2017

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.


  • Equations with measurable coefficients
  • Harnack inequality
  • Homogenization
  • Qualitative properties of solutions


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