Journal
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
Volume 35, Issue 11, Pages 2092-2122Publisher
TAYLOR & FRANCIS INC
DOI: 10.1080/03605301003735680
Keywords
Degenerate Schrodinger equation; Fractional Laplacian; Harmonic oscillator; Harnack's inequality; Heat semigroup
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Funding
- Ministerio de Ciencia e Innovacion de Espana [MTM2008-06621-C02-01]
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The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy-Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H sigma=(- + |x|2)sigma to deduce a Harnack's inequality. A pointwise formula for H sigma f(x) and some maximum and comparison principles are derived.
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