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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

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.
Citation: Mouhamed Moustapha Fall, Veronica Felli. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete & Continuous Dynamical Systems - A, 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. 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.

[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. 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. doi: 10.1080/03605300600987306.

[5]

S.-Y. A. Chang and M. d. M. Gonzàlez, Fractional Laplacian in conformal geometry,, Adv. Math., 226 (2011), 1410. 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. 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. doi: 10.1016/j.bulsci.2011.12.004.

[8]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions,, Vol. II, (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. 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. 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. 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. 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., (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. doi: 10.1002/cpa.20186.

[15]

J. Fröhlich and E. Lenzmann, Boson stars as solitary waves,, Comm. Math. Phys., 274 (2007), 1. 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. doi: 10.1512/iumj.1986.35.35015.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, 2nd edition, (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.

[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. 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. 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. doi: 10.1090/S0273-0979-1990-15831-8.

[22]

E. H. Lieb and M. Loss, Analysis,, 2nd edition, (2001). doi: 10.1090/gsm/014.

[23]

B. Opic and A. Kufner, Hardy-type Inequalities,, Pitman Research Notes in Math., (1990).

[24]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potential,, Comm. Partial Differential Equations, 40 (2015), 77. 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. doi: 10.1090/S0002-9939-2014-12594-9.

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (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. 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. 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.

[30]

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

[31]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities,, J. Funct. Anal., 168 (1999), 121. 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. 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.

[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. 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. doi: 10.1080/03605300600987306.

[5]

S.-Y. A. Chang and M. d. M. Gonzàlez, Fractional Laplacian in conformal geometry,, Adv. Math., 226 (2011), 1410. 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. 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. doi: 10.1016/j.bulsci.2011.12.004.

[8]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions,, Vol. II, (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. 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. 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. 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. 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., (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. doi: 10.1002/cpa.20186.

[15]

J. Fröhlich and E. Lenzmann, Boson stars as solitary waves,, Comm. Math. Phys., 274 (2007), 1. 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. doi: 10.1512/iumj.1986.35.35015.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, 2nd edition, (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.

[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. 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. 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. doi: 10.1090/S0273-0979-1990-15831-8.

[22]

E. H. Lieb and M. Loss, Analysis,, 2nd edition, (2001). doi: 10.1090/gsm/014.

[23]

B. Opic and A. Kufner, Hardy-type Inequalities,, Pitman Research Notes in Math., (1990).

[24]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potential,, Comm. Partial Differential Equations, 40 (2015), 77. 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. doi: 10.1090/S0002-9939-2014-12594-9.

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (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. 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. 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.

[30]

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

[31]

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

[32]

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

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