Schauder estimates for a class of non-local elliptic equations

Hongjie Dong; Doyoon Kim

doi:10.3934/dcds.2013.33.2319

Schauder estimates for a class of non-local elliptic equations

Hongjie Dong, Doyoon Kim

Research output: Contribution to journal › Article › peer-review

47 Citations (Scopus)

Abstract

We prove Schauder estimates for a class of non-local elliptic operators with kernel K(y) = a(y)/|y|^d+σ and either Dini or Hölder continuous data. Here 0 < σ < 2 is a constant and a is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz-Zygmund spaces Λ^α+σ and Λ^α for any α > 0. Several local estimates and an extension to operators with kernels K(x, y) are also discussed.

Original language	English
Pages (from-to)	2319-2347
Number of pages	29
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	33
Issue number	6
DOIs	https://doi.org/10.3934/dcds.2013.33.2319
Publication status	Published - 2013 Jun
Externally published	Yes

Keywords

Lévy processes
Non-local elliptic equations
Schauder estimates

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2013.33.2319

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author = "Hongjie Dong and Doyoon Kim",

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AU - Kim, Doyoon

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N2 - We prove Schauder estimates for a class of non-local elliptic operators with kernel K(y) = a(y)/|y|d+σ and either Dini or Hölder continuous data. Here 0 < σ < 2 is a constant and a is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz-Zygmund spaces Λα+σ and Λα for any α > 0. Several local estimates and an extension to operators with kernels K(x, y) are also discussed.

AB - We prove Schauder estimates for a class of non-local elliptic operators with kernel K(y) = a(y)/|y|d+σ and either Dini or Hölder continuous data. Here 0 < σ < 2 is a constant and a is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz-Zygmund spaces Λα+σ and Λα for any α > 0. Several local estimates and an extension to operators with kernels K(x, y) are also discussed.

KW - Lévy processes

KW - Non-local elliptic equations

KW - Schauder estimates

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M3 - Article

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SP - 2319

EP - 2347

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 6

ER -

Schauder estimates for a class of non-local elliptic equations

Abstract

Keywords

ASJC Scopus subject areas

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