On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian ( − Δ ) α / 2 , 0 < α < 2 , in a bounded C 1 , 1 domain D in R n . In particular, we introduce a new Kato class K α ( D ) and we exploit the properties of this class...

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Veröffentlicht in:Nonlinear analysis 2011-03, Vol.74 (5), p.1555-1576
Hauptverfasser: Chemmam, Rym, Mâagli, Habib, Masmoudi, Syrine
Format: Artikel
Sprache:eng
Schlagworte:
Dirichlet problem
Exact sciences and technology
Fractional Laplacian
Green function
Harmonic function
Martin boundary
Martin representation
Mathematical analysis
Mathematics
Nonlinearity
Operator theory
Operators
Ordinary differential equations
Potential theory
Sciences and techniques of general use
Symmetric stable process
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Beschreibung
Zusammenfassung:The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian ( − Δ ) α / 2 , 0 < α < 2 , in a bounded C 1 , 1 domain D in R n . In particular, we introduce a new Kato class K α ( D ) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2010.10.027