-
Previous Article
Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval
- CPAA Home
- This Issue
-
Next Article
Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities
Positive powers of the Laplacian: From hypersingular integrals to boundary value problems
| 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 |
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.
References:
| [1] |
N. Abatangelo, S. Jarohs and A. Saldaña, On the maximum principle for higher-order fractional Laplacians, preprint, arXiv: 1607.00929. |
| [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. |
| [3] |
C. Bucur and E. Valdinoci,
Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the UMI, Springer International Publishing (Switzerland), 2016. |
| [4] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
| [5] |
S. Dipierro and H.-C. Grunau,
Boggio's formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl., 196 (2017), 1327-1344.
|
| [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. |
| [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. |
| [8] |
F. Gazzola, H. -C. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems, volume 1991 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010. |
| [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.
|
| [10] |
S. G. Samko, A. A. Kilbas and O. I. Marichev,
Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [11] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
|
| [12] |
X. Tian and Q. Du,
A class of high order nonlocal operators, Arch. Ration. Mech. Anal., 222 (2016), 1521-1553.
|
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. |
| [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. |
| [3] |
C. Bucur and E. Valdinoci,
Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the UMI, Springer International Publishing (Switzerland), 2016. |
| [4] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
|
| [5] |
S. Dipierro and H.-C. Grunau,
Boggio's formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl., 196 (2017), 1327-1344.
|
| [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. |
| [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. |
| [8] |
F. Gazzola, H. -C. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems, volume 1991 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010. |
| [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.
|
| [10] |
S. G. Samko, A. A. Kilbas and O. I. Marichev,
Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [11] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
|
| [12] |
X. Tian and Q. Du,
A class of high order nonlocal operators, Arch. Ration. Mech. Anal., 222 (2016), 1521-1553.
|
| [1] |
Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851 |
| [2] |
Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062 |
| [3] |
J. Ángel Cid, Pedro J. Torres. Solvability for some boundary value problems with $\phi$-Laplacian operators. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 727-732. doi: 10.3934/dcds.2009.23.727 |
| [4] |
John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84 |
| [5] |
Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127 |
| [6] |
Mei Yu, Xia Zhang, Binlin Zhang. Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3597-3612. doi: 10.3934/cpaa.2020157 |
| [7] |
Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361 |
| [8] |
Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118 |
| [9] |
Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020 |
| [10] |
Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990 |
| [11] |
Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451 |
| [12] |
K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546 |
| [13] |
Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure and Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83 |
| [14] |
Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595 |
| [15] |
Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure and Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293 |
| [16] |
John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515 |
| [17] |
John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283 |
| [18] |
John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276 |
| [19] |
Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006 |
| [20] |
Daniele Boffi, Lucia Gastaldi, Sebastian Wolf. Higher-order time-stepping schemes for fluid-structure interaction problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3807-3830. doi: 10.3934/dcdsb.2020229 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]