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December  2015, 35(12): 5827-5867. doi: 10.3934/dcds.2015.35.5827

Unique continuation properties for relativistic Schrödinger operators with a singular potential

Mouhamed Moustapha Fall 1, and  Veronica Felli 2,

1. 

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

Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 55, 20125 Milano, Italy

Received  December 2013 Published  May 2015

Asymptotics of solutions to relativistic fractional elliptic equations with Hardy type potentials is established in this paper. As a consequence, unique continuation properties are obtained.
Keywords: Fractional elliptic equations, unique continuation., Hardy inequality.
Mathematics Subject Classification: Primary: 35R11, 35J75; Secondary: 35B4.
Citation: Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5827-5867. doi: 10.3934/dcds.2015.35.5827
References:
[1]

F. J. Almgren Jr., $Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Amer. Math. Soc., 8 (1983), 327-328. doi: 10.1090/S0273-0979-1983-15106-6.  Google Scholar

[2]

T. Byczkowski, M. Ryznar and H. Byczkowska, Bessel potentials, Green functions and exponential functionals on half-spaces, Probab. Math. Statist., 26 (2006), 155-173.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

S.-Y. A. Chang and M. d. M. Gonzàlez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[6]

Z.-Q. Chen, P. Kim and R. Song, Green function estimates for relativistic stable processes in half-space-like open sets, Stochastic Process. Appl., 121 (2011), 1148-1172. doi: 10.1016/j.spa.2011.01.004.  Google Scholar

[7]

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.  Google Scholar

[8]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.  Google Scholar

[9]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397. doi: 10.1080/03605302.2013.825918.  Google Scholar

[10]

V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, Journal of the European Mathematical Society, 13 (2011), 119-174. doi: 10.4171/JEMS/246.  Google Scholar

[11]

V. Felli, A. Ferrero and S. Terracini, On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials, Discrete Contin. Dynam. Systems, 32 (2012), 3895-3956. doi: 10.3934/dcds.2012.32.3895.  Google Scholar

[12]

V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods, Milan J. Math., 80 (2012), 203-226. doi: 10.1007/s00032-012-0174-y.  Google Scholar

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appl. Sci., published online, (2015). doi: 10.1002/mma.3438.  Google Scholar

[14]

J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705. doi: 10.1002/cpa.20186.  Google Scholar

[15]

J. Fröhlich and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.  Google Scholar

[16]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, 2nd edition, Grundlehren, 224, Springer, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[18]

I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Comm. Math. Phys., 53 (1977), 285-294.  Google Scholar

[19]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators. With an appendix by E. M. Stein, Ann. of Math. (2), 121 (1985), 463-494. doi: 10.2307/1971205.  Google Scholar

[20]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.  Google Scholar

[21]

E. H. Lieb, The stability of matter: From atoms to stars, Bull. Amer. Math. Soc. (N.S.), 22 (1990), 1-49. doi: 10.1090/S0273-0979-1990-15831-8.  Google Scholar

[22]

E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[23]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Math., Vol. 219, Longman 1990.  Google Scholar

[24]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potential, Comm. Partial Differential Equations, 40 (2015), 77-114. doi: 10.1080/03605302.2014.905594.  Google Scholar

[25]

I. Seo, Unique continuation for fractional Schrödinger operators in three and higher dimensions, Proc. Amer. Math. Soc., 143 (2015), 1661-1664. doi: 10.1090/S0002-9939-2014-12594-9.  Google Scholar

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[27]

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.  Google Scholar

[28]

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975.  Google Scholar

[29]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and a critical exponent, Adv. Diff. Eq., 1 (1996), 241-264.  Google Scholar

[30]

T. H. Wolff, A property of measures in $\mathbbR^N$ and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225-284. doi: 10.1007/BF01896975.  Google Scholar

[31]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.  Google Scholar

[32]

R. Yang, On higher order extensions for the fractional Laplacian, preprint,, , ().   Google Scholar

show all references

References:
[1]

F. J. Almgren Jr., $Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Amer. Math. Soc., 8 (1983), 327-328. doi: 10.1090/S0273-0979-1983-15106-6.  Google Scholar

[2]

T. Byczkowski, M. Ryznar and H. Byczkowska, Bessel potentials, Green functions and exponential functionals on half-spaces, Probab. Math. Statist., 26 (2006), 155-173.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

S.-Y. A. Chang and M. d. M. Gonzàlez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[6]

Z.-Q. Chen, P. Kim and R. Song, Green function estimates for relativistic stable processes in half-space-like open sets, Stochastic Process. Appl., 121 (2011), 1148-1172. doi: 10.1016/j.spa.2011.01.004.  Google Scholar

[7]

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.  Google Scholar

[8]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.  Google Scholar

[9]

M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397. doi: 10.1080/03605302.2013.825918.  Google Scholar

[10]

V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, Journal of the European Mathematical Society, 13 (2011), 119-174. doi: 10.4171/JEMS/246.  Google Scholar

[11]

V. Felli, A. Ferrero and S. Terracini, On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials, Discrete Contin. Dynam. Systems, 32 (2012), 3895-3956. doi: 10.3934/dcds.2012.32.3895.  Google Scholar

[12]

V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods, Milan J. Math., 80 (2012), 203-226. doi: 10.1007/s00032-012-0174-y.  Google Scholar

[13]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appl. Sci., published online, (2015). doi: 10.1002/mma.3438.  Google Scholar

[14]

J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705. doi: 10.1002/cpa.20186.  Google Scholar

[15]

J. Fröhlich and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.  Google Scholar

[16]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, 2nd edition, Grundlehren, 224, Springer, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[18]

I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Comm. Math. Phys., 53 (1977), 285-294.  Google Scholar

[19]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators. With an appendix by E. M. Stein, Ann. of Math. (2), 121 (1985), 463-494. doi: 10.2307/1971205.  Google Scholar

[20]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.  Google Scholar

[21]

E. H. Lieb, The stability of matter: From atoms to stars, Bull. Amer. Math. Soc. (N.S.), 22 (1990), 1-49. doi: 10.1090/S0273-0979-1990-15831-8.  Google Scholar

[22]

E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[23]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Math., Vol. 219, Longman 1990.  Google Scholar

[24]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potential, Comm. Partial Differential Equations, 40 (2015), 77-114. doi: 10.1080/03605302.2014.905594.  Google Scholar

[25]

I. Seo, Unique continuation for fractional Schrödinger operators in three and higher dimensions, Proc. Amer. Math. Soc., 143 (2015), 1661-1664. doi: 10.1090/S0002-9939-2014-12594-9.  Google Scholar

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.  Google Scholar

[27]

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.  Google Scholar

[28]

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975.  Google Scholar

[29]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and a critical exponent, Adv. Diff. Eq., 1 (1996), 241-264.  Google Scholar

[30]

T. H. Wolff, A property of measures in $\mathbbR^N$ and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225-284. doi: 10.1007/BF01896975.  Google Scholar

[31]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.  Google Scholar

[32]

R. Yang, On higher order extensions for the fractional Laplacian, preprint,, , ().   Google Scholar

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