March  2019, 39(3): 1205-1235. doi: 10.3934/dcds.2019052

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

1. 

Département de mathématique, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, 1050 Ixelles, Belgium

2. 

Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia

3. 

African Institute for Mathematical Sciences (A.I.M.S) of Senegal, KM 2, Route de Joal (Centre I.R.D. Mbour), B.P. 1418 Mbour, Sénégal

4. 

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

5. 

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

* Corresponding author: Sven Jarohs

Received  June 2018 Revised  August 2018 Published  December 2018

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

Citation: Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1205-1235. doi: 10.3934/dcds.2019052
References:
[1]

N. Abatangelo, Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.  doi: 10.3934/dcds.2015.35.5555.

[2]

N. AbatangeloS. Jarohs and A. Saldaña, Green function and Martin kernel for higher-order fractional Laplacians in balls, Nonlinear Analysis, 175 (2018), 173-190.  doi: 10.1016/j.na.2018.05.019.

[3]

N. AbatangeloS. Jarohs and A. Saldaña, Positive powers of the Laplacian: From hypersingular integrals to boundary value problems, Comm. Pure Appl. Anal., 17 (2018), 899-922.  doi: 10.3934/cpaa.2018045.

[4]

N. Abatangelo, S. Jarohs and A. Saldaña, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Contemp. Math., to appear.

[5]

N. AbatangeloS. Jarohs and A. Saldaña, On the loss of maximum principles for higher-order fractional Laplacians, Proc. Amer. Math. Soc., 146 (2018), 4823-4835.  doi: 10.1090/proc/14165.

[6]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[7]

I. BacharH. Mâagli and M. Zribi, Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space, Manuscripta Math., 113 (2004), 269-291.  doi: 10.1007/s00229-003-0410-4.

[8]

K. Bogdan, Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J., 29 (1999), 227-243. 

[9]

K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist., 20 (2000), 293-335. 

[10]

K. Bogdan and T. Żak, On Kelvin transformation, J. Theoret. Probab., 19 (2006), 89-120.  doi: 10.1007/s10959-006-0003-8.

[11]

Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.  doi: 10.1007/s002080050232.

[12]

S. Dipierro and H.-C. Grunau, Boggio's formula for fractional polyharmonic Dirichlet problem, Ann. Mat. Pura Appl. (4), 196 (2017), 1327-1344.  doi: 10.1007/s10231-016-0618-z.

[13]

J. Edenhofer, Eine Integraldarstellung polyharmonischer Funktionen in einem Halbraum, Z. Angew. Math. Mech., 57 (1977), T227-T229. 

[14]

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

[15]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.

[16]

M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Commun. Contemp. Math., 18 (2016), 1550012, 25pp. doi: 10.1142/S0219199715500121.

[17]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, volume 1991 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[18]

H. W. Gould, Combinatorial Identities, Henry W. Gould, Morgantown, W.Va., 1972.

[19]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.

[20]

G. Grubb, Green's formula and Dirichlet-to-Neumann operator for fractional order pseudodifferential operators, Comm. Partial Differential Equations, to appear.

[21]

T. Grzywny, M. Kassmann and L. Leżaj, Remarks on the nonlocal Dirichlet problem, preprint, arXiv: 1807.03676v1.

[22]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin Heidelberg New York, 1972.

[23]

G. PalatucciO. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4), 192 (2013), 673-718.  doi: 10.1007/s10231-011-0243-9.

[24]

W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.  doi: 10.1007/s00209-008-0352-3.

[25]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[26]

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.  doi: 10.3934/dcds.2015.35.2131.

[27]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[28]

Y. H. ZhangG. T. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936.  doi: 10.1016/j.jde.2015.09.032.

show all references

References:
[1]

N. Abatangelo, Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607.  doi: 10.3934/dcds.2015.35.5555.

[2]

N. AbatangeloS. Jarohs and A. Saldaña, Green function and Martin kernel for higher-order fractional Laplacians in balls, Nonlinear Analysis, 175 (2018), 173-190.  doi: 10.1016/j.na.2018.05.019.

[3]

N. AbatangeloS. Jarohs and A. Saldaña, Positive powers of the Laplacian: From hypersingular integrals to boundary value problems, Comm. Pure Appl. Anal., 17 (2018), 899-922.  doi: 10.3934/cpaa.2018045.

[4]

N. Abatangelo, S. Jarohs and A. Saldaña, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Contemp. Math., to appear.

[5]

N. AbatangeloS. Jarohs and A. Saldaña, On the loss of maximum principles for higher-order fractional Laplacians, Proc. Amer. Math. Soc., 146 (2018), 4823-4835.  doi: 10.1090/proc/14165.

[6]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.

[7]

I. BacharH. Mâagli and M. Zribi, Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space, Manuscripta Math., 113 (2004), 269-291.  doi: 10.1007/s00229-003-0410-4.

[8]

K. Bogdan, Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J., 29 (1999), 227-243. 

[9]

K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist., 20 (2000), 293-335. 

[10]

K. Bogdan and T. Żak, On Kelvin transformation, J. Theoret. Probab., 19 (2006), 89-120.  doi: 10.1007/s10959-006-0003-8.

[11]

Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501.  doi: 10.1007/s002080050232.

[12]

S. Dipierro and H.-C. Grunau, Boggio's formula for fractional polyharmonic Dirichlet problem, Ann. Mat. Pura Appl. (4), 196 (2017), 1327-1344.  doi: 10.1007/s10231-016-0618-z.

[13]

J. Edenhofer, Eine Integraldarstellung polyharmonischer Funktionen in einem Halbraum, Z. Angew. Math. Mech., 57 (1977), T227-T229. 

[14]

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

[15]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.

[16]

M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Commun. Contemp. Math., 18 (2016), 1550012, 25pp. doi: 10.1142/S0219199715500121.

[17]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, volume 1991 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[18]

H. W. Gould, Combinatorial Identities, Henry W. Gould, Morgantown, W.Va., 1972.

[19]

G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.

[20]

G. Grubb, Green's formula and Dirichlet-to-Neumann operator for fractional order pseudodifferential operators, Comm. Partial Differential Equations, to appear.

[21]

T. Grzywny, M. Kassmann and L. Leżaj, Remarks on the nonlocal Dirichlet problem, preprint, arXiv: 1807.03676v1.

[22]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin Heidelberg New York, 1972.

[23]

G. PalatucciO. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4), 192 (2013), 673-718.  doi: 10.1007/s10231-011-0243-9.

[24]

W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.  doi: 10.1007/s00209-008-0352-3.

[25]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[26]

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.  doi: 10.3934/dcds.2015.35.2131.

[27]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[28]

Y. H. ZhangG. T. Deng and T. Qian, Integral representations of a class of harmonic functions in the half space, J. Differential Equations, 260 (2016), 923-936.  doi: 10.1016/j.jde.2015.09.032.

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