Harmonic functions for a class of integro-differential operators

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

LanguageEnglish
Pages21-44
Number of pages24
JournalPotential Analysis
Volume31
Issue number1
Early online date6 Mar 2009
DOIs
Publication statusPublished - 31 Jul 2009
Externally publishedYes

Fingerprint

Integro-differential Operators
Harnack Inequality
Harmonic Functions
Non-negative
Operator
Harmonic
Regularity
Theorem
Class

Keywords

  • harmonic functions
  • Harnack inequality
  • integro-differential operators
  • jump processes

Cite this

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title = "Harmonic functions for a class of integro-differential operators",
abstract = "We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.",
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author = "Mohammud Foondun",
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Harmonic functions for a class of integro-differential operators. / Foondun, Mohammud.

In: Potential Analysis, Vol. 31, No. 1, 31.07.2009, p. 21-44.

Research output: Contribution to journalArticle

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N2 - We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

AB - We consider the operator LL defined on C2(Rd)C2(Rd) functions by Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Lf(x)=12∑i,j=1daij(x)∂2f(x)∂xi∂xj+∑i=1dbi(x)∂f(x)∂xi+∫Rd∖{0}[f(x+h)−f(x)−1(|h|≤1)h⋅∇f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in RdRd and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

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KW - Harnack inequality

KW - integro-differential operators

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