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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[32]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[32]

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

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