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Optimal actuator location of the minimum norm controls for stochastic heat equations
Quantitative unique continuation for the heat equation with Coulomb potentials
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 |
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.
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Observation estimate for kinetic transport equation by diffusion approximation, Comptes Rendus Mathematique, 355 (2017), 640-664.
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A strong unique continuation theorem for parabolic equations, Mathematische Annalen, 311 (1998), 603-630.
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Control and stabilization properties for a singular heat equation with an inverse-square potential, Communications in Partial Differential Equations, 33 (2008), 1996-2019.
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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.
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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.
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L. Escauriaza and F. J. Fernández,
Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.
doi: 10.1007/BF02384566. |
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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: Ap 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: Ap 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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