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Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions
Fundamental solutions of a class of homogeneous integro-differential elliptic equations
1. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China |
2. | Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China |
3. | Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA |
In this paper, we study a class of integro-differential elliptic operators $L_{σ}$ with kernel $k(y) = a(y)/|y|^{d+σ}$, where $d≥2, σ∈(0,2)$, and the positive function $a(y)$ is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution $Φ$ of $L_{σ}$ if $a(y)$ satisfies a natural cancellation assumption and $|a(y)-1|$ is small. Furthermore, we show that the fundamental solution $Φ$ is $-α^{*}$ homogeneous and Lipschitz continuous, where the constant $α^{*}∈(0,d)$. A Liouville-type theorem demonstrates that the fundamental solution $Φ$ is the unique nontrivial solution of $L_{σ}u = 0$ in $\mathbb{R}^{d}\setminus\{0\}$ that is bounded from below.
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W. Ao, J. Wei and W. Yang,
Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dynami. Systems, 37 (2017), 5561-5601.
doi: 10.3934/dcds.2017242. |
[2] |
S. N. Armstrong, B. Sirakov and C. K. Smart,
Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math., 64 (2011), 737-777.
doi: 10.1002/cpa.20360. |
[3] |
S. N. Armstrong, B. Sirakov and C. K. Smart,
Singular solutions of fully nonlinear elliptic equations and applications, Arch. Rational Mech. Anal., 205 (2013), 345-394.
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[4] |
R. F. Bass and M. Kassmann,
Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., 357 (2005), 837-850.
doi: 10.1090/S0002-9947-04-03549-4. |
[5] |
M. Bôcher,
Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc., 9 (1903), 455-465.
doi: 10.1090/S0002-9904-1903-01017-9. |
[6] |
K. Bogdan, T. Kulczycki and A. Nowak,
Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.
|
[7] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. Poincaré, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[8] |
L. Caffarelli,
Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213.
doi: 10.2307/1971480. |
[9] |
L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, Vol. 43, Providence, R.I., 1995.
doi: 10.1090/coll/043. |
[10] |
L. Caffarelli, Y. Y. Li and L. Nirenberg,
Some remarks on singular solutions of nonlinear elliptic equations. Ⅰ, J. Fixed point Theory Appl., 5 (2009), 353-395.
doi: 10.1007/s11784-009-0107-8. |
[11] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[12] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[13] |
L. Caffarelli and L. Silvestre,
The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math., 174 (2011), 1163-1187.
doi: 10.4007/annals.2011.174.2.9. |
[14] |
W. Chen, L. D'Ambrosio and Y. Li,
Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381.
doi: 10.1016/j.na.2014.11.003. |
[15] |
Y. Chen and C. Wei,
Partial regularity of solutions to the fractional Navier-Stokes equations, Discrete Contin. Dynami. Systems, 36 (2016), 5309-5322.
doi: 10.3934/dcds.2016033. |
[16] |
Z. Chen and X. Zhang,
Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields, 165 (2016), 267-312.
doi: 10.1007/s00440-015-0631-y. |
[17] |
T. Cheng,
Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dynami. Systems, 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[18] |
H. Dong and D. Kim,
On $ L^{p}$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199.
doi: 10.1016/j.jfa.2011.11.002. |
[19] |
H. Dong and D. Kim,
Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dynami. Systems, 33 (2013), 2319-2347.
doi: 10.3934/dcds.2013.33.2319. |
[20] |
P. Felmer and A. Quaas,
Fundamental solutions for a class of Isaacs integral operators, Discrete Contin. Dynami. Systems, 30 (2011), 493-508.
doi: 10.3934/dcds.2011.30.493. |
[21] |
P. Felmer and A. Quaas,
Fundamental solutions and Liouville type theorems for nonlinear operators, Adv. Math., 226 (2011), 2712-2738.
doi: 10.1016/j.aim.2010.09.023. |
[22] |
D. Gilbarg and J. Serrin,
On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math., 4 (1955/56), 309-340.
doi: 10.1007/BF02787726. |
[23] |
D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[24] |
D. Labutin,
Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214.
doi: 10.1007/s002050000108. |
[25] |
D. Labutin,
Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations, 177 (2001), 49-76.
doi: 10.1006/jdeq.2001.3998. |
[26] |
L. Silvestre,
Hölder estimates for solutions of integro differential equations like the fractional laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.
doi: 10.1512/iumj.2006.55.2706. |
[27] |
E. M. Stein, Singular Integrals and Differential Property of Functions, Princeton press, Princeton, 1970. |
[28] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[29] |
L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman research notes in mathematics Series, 353. Longman. Harlow, 1996. |
[30] |
Z. Wang and H. Zhou,
Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dynami. Systems, 36 (2016), 499-508.
doi: 10.3934/dcds.2016.36.499. |
[31] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
A Liouville theorem for the fractional Laplacian, Mathematics, 2 (2014), 423-452.
|
[32] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dynami. Systems, 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
show all references
References:
[1] |
W. Ao, J. Wei and W. Yang,
Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials, Discrete Contin. Dynami. Systems, 37 (2017), 5561-5601.
doi: 10.3934/dcds.2017242. |
[2] |
S. N. Armstrong, B. Sirakov and C. K. Smart,
Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math., 64 (2011), 737-777.
doi: 10.1002/cpa.20360. |
[3] |
S. N. Armstrong, B. Sirakov and C. K. Smart,
Singular solutions of fully nonlinear elliptic equations and applications, Arch. Rational Mech. Anal., 205 (2013), 345-394.
doi: 10.1007/s00205-012-0505-8. |
[4] |
R. F. Bass and M. Kassmann,
Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., 357 (2005), 837-850.
doi: 10.1090/S0002-9947-04-03549-4. |
[5] |
M. Bôcher,
Singular points of functions which satisfy partial differential equations of the elliptic type, Bull. Amer. Math. Soc., 9 (1903), 455-465.
doi: 10.1090/S0002-9904-1903-01017-9. |
[6] |
K. Bogdan, T. Kulczycki and A. Nowak,
Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556.
|
[7] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. I. Poincaré, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[8] |
L. Caffarelli,
Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213.
doi: 10.2307/1971480. |
[9] |
L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, Vol. 43, Providence, R.I., 1995.
doi: 10.1090/coll/043. |
[10] |
L. Caffarelli, Y. Y. Li and L. Nirenberg,
Some remarks on singular solutions of nonlinear elliptic equations. Ⅰ, J. Fixed point Theory Appl., 5 (2009), 353-395.
doi: 10.1007/s11784-009-0107-8. |
[11] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[12] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[13] |
L. Caffarelli and L. Silvestre,
The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math., 174 (2011), 1163-1187.
doi: 10.4007/annals.2011.174.2.9. |
[14] |
W. Chen, L. D'Ambrosio and Y. Li,
Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381.
doi: 10.1016/j.na.2014.11.003. |
[15] |
Y. Chen and C. Wei,
Partial regularity of solutions to the fractional Navier-Stokes equations, Discrete Contin. Dynami. Systems, 36 (2016), 5309-5322.
doi: 10.3934/dcds.2016033. |
[16] |
Z. Chen and X. Zhang,
Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Relat. Fields, 165 (2016), 267-312.
doi: 10.1007/s00440-015-0631-y. |
[17] |
T. Cheng,
Monotonicity and symmetry of solutions to fractional Laplacian equation, Discrete Contin. Dynami. Systems, 37 (2017), 3587-3599.
doi: 10.3934/dcds.2017154. |
[18] |
H. Dong and D. Kim,
On $ L^{p}$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199.
doi: 10.1016/j.jfa.2011.11.002. |
[19] |
H. Dong and D. Kim,
Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dynami. Systems, 33 (2013), 2319-2347.
doi: 10.3934/dcds.2013.33.2319. |
[20] |
P. Felmer and A. Quaas,
Fundamental solutions for a class of Isaacs integral operators, Discrete Contin. Dynami. Systems, 30 (2011), 493-508.
doi: 10.3934/dcds.2011.30.493. |
[21] |
P. Felmer and A. Quaas,
Fundamental solutions and Liouville type theorems for nonlinear operators, Adv. Math., 226 (2011), 2712-2738.
doi: 10.1016/j.aim.2010.09.023. |
[22] |
D. Gilbarg and J. Serrin,
On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math., 4 (1955/56), 309-340.
doi: 10.1007/BF02787726. |
[23] |
D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[24] |
D. Labutin,
Removable singularities for fully nonlinear elliptic equations, Arch. Rational Mech. Anal., 155 (2000), 201-214.
doi: 10.1007/s002050000108. |
[25] |
D. Labutin,
Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations, 177 (2001), 49-76.
doi: 10.1006/jdeq.2001.3998. |
[26] |
L. Silvestre,
Hölder estimates for solutions of integro differential equations like the fractional laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.
doi: 10.1512/iumj.2006.55.2706. |
[27] |
E. M. Stein, Singular Integrals and Differential Property of Functions, Princeton press, Princeton, 1970. |
[28] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[29] |
L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman research notes in mathematics Series, 353. Longman. Harlow, 1996. |
[30] |
Z. Wang and H. Zhou,
Radial sign-changing solution for fractional Schrödinger equation, Discrete Contin. Dynami. Systems, 36 (2016), 499-508.
doi: 10.3934/dcds.2016.36.499. |
[31] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
A Liouville theorem for the fractional Laplacian, Mathematics, 2 (2014), 423-452.
|
[32] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dynami. Systems, 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
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