On second order elliptic and parabolic equations of mixed type

Research output: Contribution to journalArticle

Abstract

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
Volume272
Issue number8
DOIs
StatePublished - Apr 15 2017

Fingerprint

Harnack Inequality
Second Order Elliptic Equations
Elliptic Equations
Parabolic Equation
Strictly positive
Homogenization
Divergence
Interior
Bibliography
Form

Keywords

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

Cite this

On second order elliptic and parabolic equations of mixed type. / Chen, Gong; Safonov, Mikhail V.

In: Journal of Functional Analysis, Vol. 272, No. 8, 15.04.2017, p. 3216-3237.

Research output: Contribution to journalArticle

@article{625e4891ac00454dbd2c94c6b6e96916,
title = "On second order elliptic and parabolic equations of mixed type",
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∈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 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: 22 titles.",
keywords = "Equations with measurable coefficients, Harnack inequality, Homogenization, Qualitative properties of solutions",
author = "Gong Chen and Safonov, {Mikhail V}",
year = "2017",
month = "4",
day = "15",
doi = "10.1016/j.jfa.2016.12.027",
language = "English (US)",
volume = "272",
pages = "3216--3237",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "8",

}

TY - JOUR

T1 - On second order elliptic and parabolic equations of mixed type

AU - Chen, Gong

AU - Safonov, Mikhail V

PY - 2017/4/15

Y1 - 2017/4/15

N2 - 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.

AB - 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.

KW - Equations with measurable coefficients

KW - Harnack inequality

KW - Homogenization

KW - Qualitative properties of solutions

UR - http://www.scopus.com/inward/record.url?scp=85008616153&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85008616153&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2016.12.027

DO - 10.1016/j.jfa.2016.12.027

M3 - Article

VL - 272

SP - 3216

EP - 3237

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -