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Non-local sublinear problems: Existence, comparison, and radial symmetry
Department of Mathematics and Informatics, via Ospedale 72, 09124 Cagliari, Italy |
We establish a symmetry result for a non-autonomous overdetermined problem associated to a sublinear fractional equation. To this purpose we prove, in particular, that the solution of the corresponding Dirichlet problem is monotonically increasing with respect to the domain. We also obtain a strong minimum principle and a boundary-point lemma for linear fractional equations that may have an independent interest.
References:
[1] |
H. Brézis and L. Oswald,
Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.
doi: 10.1016/0362-546X(86)90011-8. |
[2] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[3] |
L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[4] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[5] |
J. I. Díaz and J. E. Saá,
Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.
|
[6] |
A. Erdélyi (Editor), Higher Transcendental Functions, McGraw-Hill, 1953. |
[7] |
M. M. Fall and S. Jarohs,
Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21 (2015), 924-938.
doi: 10.1051/cocv/2014048. |
[8] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Annales Academiae Scientiarum Fennicae Mathematica, 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[9] |
R. K. Getoor,
First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.
doi: 10.1090/S0002-9947-1961-0137148-5. |
[10] |
A. Greco,
A characterization of the ellipsoid through the torsion problem, J. Appl. Math. Phys. (ZAMP), 59 (2008), 753-765.
doi: 10.1007/s00033-007-7040-8. |
[11] |
A. Greco,
Boundary point lemmas and overdetermined problems, J. Math. Anal. Appl., 278 (2003), 214-224.
doi: 10.1016/S0022-247X(02)00656-X. |
[12] |
A. Greco,
Comparison principle and constrained radial symmetry for the subdiffusive p-Laplacian, Publ. Mat., 58 (2014), 485-498.
doi: 10.5565/PUBLMAT_58214_24. |
[13] |
A. Greco,
Constrained radial symmetry for the infinity-Laplacian, Nonlinear Analysis: Real World Applications, 37 (2017), 239-248.
doi: 10.1016/j.nonrwa.2017.02.016. |
[14] |
A. Greco,
Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387-399.
|
[15] |
A. Greco and R. Servadei,
Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.
doi: 10.4310/MRL.2016.v23.n3.a14. |
[16] |
A. Iannizzotto, S. Mosconi and M. Squassina,
Hs versus C0-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.
doi: 10.1007/s00030-014-0292-z. |
[17] |
S. Jarohs and T. Weth, On the strong maximum principle for nonlocal operators, preprint, arXiv: 1702.08767. |
[18] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[19] |
V. Mascia, Un Problema Sublineare Non Locale (Italian), Thesis. University of Cagliari, 2017. |
[20] |
G. Molica Bisci and V. D. Rădulescu,
Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl., 22 (2015), 721-739.
doi: 10.1007/s00030-014-0302-1. |
[21] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[22] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[23] |
E. M. Stein,
The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67 (1961), 102-104.
doi: 10.1090/S0002-9904-1961-10517-X. |
[24] |
R. L. Wheeden,
On hypersingular integrals and Lebesgue spaces of differentiable functions, Trans. Amer. Math. Soc., 134 (1968), 421-435.
doi: 10.1090/S0002-9947-1968-0232249-1. |
show all references
References:
[1] |
H. Brézis and L. Oswald,
Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55-64.
doi: 10.1016/0362-546X(86)90011-8. |
[2] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[3] |
L. Cesari, Optimization - Theory and Applications, Springer-Verlag, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[4] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[5] |
J. I. Díaz and J. E. Saá,
Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.
|
[6] |
A. Erdélyi (Editor), Higher Transcendental Functions, McGraw-Hill, 1953. |
[7] |
M. M. Fall and S. Jarohs,
Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21 (2015), 924-938.
doi: 10.1051/cocv/2014048. |
[8] |
A. Fiscella, R. Servadei and E. Valdinoci,
Density properties for fractional Sobolev spaces, Annales Academiae Scientiarum Fennicae Mathematica, 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[9] |
R. K. Getoor,
First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.
doi: 10.1090/S0002-9947-1961-0137148-5. |
[10] |
A. Greco,
A characterization of the ellipsoid through the torsion problem, J. Appl. Math. Phys. (ZAMP), 59 (2008), 753-765.
doi: 10.1007/s00033-007-7040-8. |
[11] |
A. Greco,
Boundary point lemmas and overdetermined problems, J. Math. Anal. Appl., 278 (2003), 214-224.
doi: 10.1016/S0022-247X(02)00656-X. |
[12] |
A. Greco,
Comparison principle and constrained radial symmetry for the subdiffusive p-Laplacian, Publ. Mat., 58 (2014), 485-498.
doi: 10.5565/PUBLMAT_58214_24. |
[13] |
A. Greco,
Constrained radial symmetry for the infinity-Laplacian, Nonlinear Analysis: Real World Applications, 37 (2017), 239-248.
doi: 10.1016/j.nonrwa.2017.02.016. |
[14] |
A. Greco,
Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387-399.
|
[15] |
A. Greco and R. Servadei,
Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.
doi: 10.4310/MRL.2016.v23.n3.a14. |
[16] |
A. Iannizzotto, S. Mosconi and M. Squassina,
Hs versus C0-weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497.
doi: 10.1007/s00030-014-0292-z. |
[17] |
S. Jarohs and T. Weth, On the strong maximum principle for nonlocal operators, preprint, arXiv: 1702.08767. |
[18] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[19] |
V. Mascia, Un Problema Sublineare Non Locale (Italian), Thesis. University of Cagliari, 2017. |
[20] |
G. Molica Bisci and V. D. Rădulescu,
Multiplicity results for elliptic fractional equations with subcritical term, Nonlinear Differ. Equ. Appl., 22 (2015), 721-739.
doi: 10.1007/s00030-014-0302-1. |
[21] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[22] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[23] |
E. M. Stein,
The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67 (1961), 102-104.
doi: 10.1090/S0002-9904-1961-10517-X. |
[24] |
R. L. Wheeden,
On hypersingular integrals and Lebesgue spaces of differentiable functions, Trans. Amer. Math. Soc., 134 (1968), 421-435.
doi: 10.1090/S0002-9947-1968-0232249-1. |
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