Quantitative unique continuation for the heat equation with Coulomb potentials

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September & December 2018, 8(3&4): 1097-1116. doi: 10.3934/mcrf.2018047

Quantitative unique continuation for the heat equation with Coulomb potentials

Can Zhang ^1,2,,

1.	School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China
2.	Department of Mathematics, University of the Basque Country (UPV/EHU), 48940 Leioa, Bilbao, Spain

* Corresponding author: Can Zhang

Received July 2017 Revised September 2017 Published September 2018

Fund Project: The author is supported by the National Natural Science Foundation of China under grants 11501424, and by Ministerio de Ciencia e Innovación grant MTM2014-53145-P, Spain.

In this paper, we establish a Hölder-type quantitative estimate of unique continuation for solutions to the heat equation with Coulomb potentials in either a bounded convex domain or a $C^2$-smooth bounded domain. The approach is based on the frequency function method, as well as some parabolic-type Hardy inequalities.

Keywords: Unique continuation, frequency function method, parabolic-type Hardy inequalities.

Mathematics Subject Classification: Primary: 93B07, 93C25.

Citation: Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047

References:

[1]	J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475. doi: 10.4171/JEMS/490.
[2]	C. Bardos and K. D. Phung, Observation estimate for kinetic transport equation by diffusion approximation, Comptes Rendus Mathematique, 355 (2017), 640-664. doi: 10.1016/j.crma.2017.04.017.
[3]	X. Y. Chen, A strong unique continuation theorem for parabolic equations, Mathematische Annalen, 311 (1998), 603-630. doi: 10.1007/s002080050202.
[4]	S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Communications in Partial Differential Equations, 33 (2008), 1996-2019. doi: 10.1080/03605300802402633.
[5]	L. Escauriaza, S. Montaner and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), 837-867. doi: 10.1016/j.matpur.2015.05.005.
[6]	L. Escauriaza, S. Montaner and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM Journal on Mathematical Analysis, 49 (2017), 4064-4092. doi: 10.1137/15M1039705.
[7]	L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127. doi: 10.1215/S0012-7094-00-10415-2.
[8]	L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60. doi: 10.1007/BF02384566.
[9]	L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.
[10]	L. C. Evans, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, American Mathematical Soc., 2010. doi: 10.1090/gsm/019.
[11]	V. Felli and A. Primo, Classification of local asymptotic for solutions to heat equations with inverse-square potentials, Discrete and Continuous Dynamical Systems - A, 31 (2011), 65-107. doi: 10.3934/dcds.2011.31.65.
[12]	F. J. Fernández, Unique continuation for parabolic operators Ⅱ, Communications in Partial Differential Equations, 28 (2003), 1597-1604. doi: 10.1081/PDE-120024523.
[13]	N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals: A_p weights and unique continuation, Indiana University Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.
[14]	N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: A geometric-variation approach, Comm. Pure. Appl. Math, 40 (1987), 347-366. doi: 10.1002/cpa.3160400305.
[15]	I. Kukavica and K. Nyström, Unique continuation on the boundary for Dini domains, Proc. Amer. Math. Soc., 126 (1998), 441-446. doi: 10.1090/S0002-9939-98-04065-9.
[16]	X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.
[17]	F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure. Appl. Math, 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.
[18]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin Heildeberg New York, 1971.
[19]	Q. Lü and Z. Yin, Unique continuation for stochastic heat equations, ESAIM Control Optim. Calc. Var., 21 (2015), 378-398. doi: 10.1051/cocv/2014027.
[20]	Q. Lü, Strong unique continuation property for stochastic parabolic equations, preprint, arXiv: 1701.02136.
[21]	T. Okaji, A note on unique continuation for parabolic operators with singular potentials, in Studies in Phase Space Analysis with Applications to PDEs, Springer New York, 84 (2013), 291-312. doi: 10.1007/978-1-4614-6348-1_13.
[22]	K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, preprint.
[23]	K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247. doi: 10.1016/j.jfa.2010.04.015.
[24]	K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703. doi: 10.4171/JEMS/371.
[25]	K. D. Phung, G. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, Journal of Differential Equations, 263 (2017), 5012-5041. doi: 10.1016/j.jde.2017.06.008.
[26]	K. D. Phung, L. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Annales de I'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 477-499. doi: 10.1016/j.anihpc.2013.04.005.
[27]	C. C. Poon, Unique continuation for parabolic equations, Communications in Partial Differential Equations, 21 (1996), 521-539. doi: 10.1080/03605309608821195.
[28]	J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902. doi: 10.1016/j.jfa.2007.12.015.
[29]	J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.
[30]	S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.
[31]	G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886. doi: 10.1137/15M1051907.
[32]	G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958. doi: 10.1137/110857398.
[33]	L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964. doi: 10.1137/140997452.
[34]	H. Yu, Approximation of time optimal controls for heat equations with perturbations in the system potential, SIAM J. Control Optim., 52 (2014), 1663-1692. doi: 10.1137/120904251.
[35]	X. Zhang, Unique continuation for stochastic parabolic equations, Differential Integral Equations, 21 (2008), 81-93.
[36]	Y. Zhang, Two equivalence theorems of different kinds of optimal control problems for Schrödinger equations, SIAM J. Control Optim., 53 (2015), 926-947. doi: 10.1137/130941195.
[37]	Y. Zhang, Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, 354 (2016), 389-393. doi: 10.1016/j.crma.2016.01.009.

show all references

References:

[1]	J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475. doi: 10.4171/JEMS/490.
[2]	C. Bardos and K. D. Phung, Observation estimate for kinetic transport equation by diffusion approximation, Comptes Rendus Mathematique, 355 (2017), 640-664. doi: 10.1016/j.crma.2017.04.017.
[3]	X. Y. Chen, A strong unique continuation theorem for parabolic equations, Mathematische Annalen, 311 (1998), 603-630. doi: 10.1007/s002080050202.
[4]	S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Communications in Partial Differential Equations, 33 (2008), 1996-2019. doi: 10.1080/03605300802402633.
[5]	L. Escauriaza, S. Montaner and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), 837-867. doi: 10.1016/j.matpur.2015.05.005.
[6]	L. Escauriaza, S. Montaner and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM Journal on Mathematical Analysis, 49 (2017), 4064-4092. doi: 10.1137/15M1039705.
[7]	L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127. doi: 10.1215/S0012-7094-00-10415-2.
[8]	L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60. doi: 10.1007/BF02384566.
[9]	L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.
[10]	L. C. Evans, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, American Mathematical Soc., 2010. doi: 10.1090/gsm/019.
[11]	V. Felli and A. Primo, Classification of local asymptotic for solutions to heat equations with inverse-square potentials, Discrete and Continuous Dynamical Systems - A, 31 (2011), 65-107. doi: 10.3934/dcds.2011.31.65.
[12]	F. J. Fernández, Unique continuation for parabolic operators Ⅱ, Communications in Partial Differential Equations, 28 (2003), 1597-1604. doi: 10.1081/PDE-120024523.
[13]	N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals: A_p weights and unique continuation, Indiana University Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.
[14]	N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: A geometric-variation approach, Comm. Pure. Appl. Math, 40 (1987), 347-366. doi: 10.1002/cpa.3160400305.
[15]	I. Kukavica and K. Nyström, Unique continuation on the boundary for Dini domains, Proc. Amer. Math. Soc., 126 (1998), 441-446. doi: 10.1090/S0002-9939-98-04065-9.
[16]	X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.
[17]	F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure. Appl. Math, 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.
[18]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin Heildeberg New York, 1971.
[19]	Q. Lü and Z. Yin, Unique continuation for stochastic heat equations, ESAIM Control Optim. Calc. Var., 21 (2015), 378-398. doi: 10.1051/cocv/2014027.
[20]	Q. Lü, Strong unique continuation property for stochastic parabolic equations, preprint, arXiv: 1701.02136.
[21]	T. Okaji, A note on unique continuation for parabolic operators with singular potentials, in Studies in Phase Space Analysis with Applications to PDEs, Springer New York, 84 (2013), 291-312. doi: 10.1007/978-1-4614-6348-1_13.
[22]	K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, preprint.
[23]	K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247. doi: 10.1016/j.jfa.2010.04.015.
[24]	K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703. doi: 10.4171/JEMS/371.
[25]	K. D. Phung, G. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, Journal of Differential Equations, 263 (2017), 5012-5041. doi: 10.1016/j.jde.2017.06.008.
[26]	K. D. Phung, L. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Annales de I'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 477-499. doi: 10.1016/j.anihpc.2013.04.005.
[27]	C. C. Poon, Unique continuation for parabolic equations, Communications in Partial Differential Equations, 21 (1996), 521-539. doi: 10.1080/03605309608821195.
[28]	J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902. doi: 10.1016/j.jfa.2007.12.015.
[29]	J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.
[30]	S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.
[31]	G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886. doi: 10.1137/15M1051907.
[32]	G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958. doi: 10.1137/110857398.
[33]	L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964. doi: 10.1137/140997452.
[34]	H. Yu, Approximation of time optimal controls for heat equations with perturbations in the system potential, SIAM J. Control Optim., 52 (2014), 1663-1692. doi: 10.1137/120904251.
[35]	X. Zhang, Unique continuation for stochastic parabolic equations, Differential Integral Equations, 21 (2008), 81-93.
[36]	Y. Zhang, Two equivalence theorems of different kinds of optimal control problems for Schrödinger equations, SIAM J. Control Optim., 53 (2015), 926-947. doi: 10.1137/130941195.
[37]	Y. Zhang, Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, 354 (2016), 389-393. doi: 10.1016/j.crma.2016.01.009.

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