October  2015, 35(10): 4905-4929. doi: 10.3934/dcds.2015.35.4905

Wave extension problem for the fractional Laplacian

1. 

Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland

2. 

Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden

3. 

Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT, 28049 Madrid, Spain

Received  November 2014 Revised  February 2015 Published  April 2015

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.
Citation: Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
References:
[1]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306.

[2]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998).

[3]

J. E. Galé, P. J. Miana and P. R. Stinga, Extension problem and fractional operators: Semigroups and wave equations,, J. Evol. Equ., 13 (2013), 343. doi: 10.1007/s00028-013-0182-6.

[4]

N. N. Lebedev, Special Functions and Their Applications,, Revised English edition, (1965).

[5]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. Partial Differential Equations, 35 (2010), 2092. doi: 10.1080/03605301003735680.

[6]

R. S. Strichartz, Convolutions with kernels having singularities on a sphere,, Trans. Amer. Math. Soc., 148 (1970), 461. doi: 10.1090/S0002-9947-1970-0256219-1.

show all references

References:
[1]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306.

[2]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998).

[3]

J. E. Galé, P. J. Miana and P. R. Stinga, Extension problem and fractional operators: Semigroups and wave equations,, J. Evol. Equ., 13 (2013), 343. doi: 10.1007/s00028-013-0182-6.

[4]

N. N. Lebedev, Special Functions and Their Applications,, Revised English edition, (1965).

[5]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. Partial Differential Equations, 35 (2010), 2092. doi: 10.1080/03605301003735680.

[6]

R. S. Strichartz, Convolutions with kernels having singularities on a sphere,, Trans. Amer. Math. Soc., 148 (1970), 461. doi: 10.1090/S0002-9947-1970-0256219-1.

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