Wave extension problem for the fractional Laplacian

Mikko Kemppainen; Peter Sjögren; José Luis Torrea

doi:10.3934/dcds.2015.35.4905

Article Contents

2015, Volume 35, Issue 10: 4905-4929. Doi: 10.3934/dcds.2015.35.4905

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Wave extension problem for the fractional Laplacian

1.
Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki
2.
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT, 28049 Madrid

Received: October 31, 2014

Revised: January 31, 2015

Published: September 2015

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Abstract

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension problem is obtained from the Schrödinger group by means of an oscillatory subordination formula, which also allows us to find kernel representations for such solutions. Asymptotics of related oscillatory integrals are analysed in order to determine the correct domains for initial data in the general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions is also described.

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Mathematics Subject Classification: Primary: 35L80; Secondary: 35C15, 33C10.

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References

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