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May  2018, 17(3): 899-922. doi: 10.3934/cpaa.2018045

Positive powers of the Laplacian: From hypersingular integrals to boundary value problems

Nicola Abatangelo 1, , Sven Jarohs 2, and  Alberto Saldaña 3,

1. 

Département de Mathématique, Université Libre de Bruxelles, boulevard du Triomphe CP 214, 1050 Bruxelles, Belgium
2. 

Institut für Mathematik, Goethe-Universität, Robert-Mayer-Straße 10, 60054 Frankfurt, Germany
3. 

Institut für Analysis, Karlsruher Institut für Technologie, Englerstraße 2, 76131 Karlsruhe, Germany

Received  September 2017 Revised  September 2017 Published  January 2018

Fund Project: The third author was supported by a research fellowship from the Alexander von Humboldt Foundation.

Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-contained as possible and relies on elementary pointwise calculations and known identities for special functions.

Keywords: Fractional Laplacian, hypersingular kernels, higher-order operators, polylaplacian, Fourier symbol, boundary value problems.
Mathematics Subject Classification: Primary: 35J40, 47N20, 26A33; Secondary: 47G30.
Citation: Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian: From hypersingular integrals to boundary value problems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 899-922. doi: 10.3934/cpaa.2018045
References:
[1]

N. Abatangelo, S. Jarohs and A. Saldaña, On the maximum principle for higher-order fractional Laplacians, preprint, arXiv: 1607.00929. Google Scholar

[2]

N. Abatangelo, S. Jarohs and A. Saldaña, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, to appear on Comm. Cont. Math., preprint at arXiv: 1707.03603. Google Scholar

[3]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the UMI, Springer International Publishing (Switzerland), 2016.  Google Scholar

[4]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.   Google Scholar

[5]

S. Dipierro and H.-C. Grunau, Boggio's formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl., 196 (2017), 1327-1344.   Google Scholar

[6]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954.  Google Scholar

[7]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.  Google Scholar

[8]

F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, volume 1991 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.  Google Scholar

[9]

X. Ros-Oton and J. Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst., 35 (2015), 2131-2150.   Google Scholar

[10]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[11]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.   Google Scholar

[12]

X. Tian and Q. Du, A class of high order nonlocal operators, Arch. Ration. Mech. Anal., 222 (2016), 1521-1553.   Google Scholar

show all references

References:
[1]

N. Abatangelo, S. Jarohs and A. Saldaña, On the maximum principle for higher-order fractional Laplacians, preprint, arXiv: 1607.00929. Google Scholar

[2]

N. Abatangelo, S. Jarohs and A. Saldaña, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, to appear on Comm. Cont. Math., preprint at arXiv: 1707.03603. Google Scholar

[3]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the UMI, Springer International Publishing (Switzerland), 2016.  Google Scholar

[4]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.   Google Scholar

[5]

S. Dipierro and H.-C. Grunau, Boggio's formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl., 196 (2017), 1327-1344.   Google Scholar

[6]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954.  Google Scholar

[7]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.  Google Scholar

[8]

F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, volume 1991 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.  Google Scholar

[9]

X. Ros-Oton and J. Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst., 35 (2015), 2131-2150.   Google Scholar

[10]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[11]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.   Google Scholar

[12]

X. Tian and Q. Du, A class of high order nonlocal operators, Arch. Ration. Mech. Anal., 222 (2016), 1521-1553.   Google Scholar

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