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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, 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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