September  2018, 8(3&4): 899-933. doi: 10.3934/mcrf.2018040

Carleman commutator approach in logarithmic convexity for parabolic equations

Institut Denis Poisson, CNRS, UMR 7013, Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France

Received  August 2017 Revised  December 2017 Published  September 2018

Fund Project: This work is supported by the Région Centre (France) - THESPEGE Project.

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.

Citation: Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
References:
[1]

S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach Space, Comm. Pure Appl. Math., 16 (1963), 121-239.  doi: 10.1002/cpa.3160160204.  Google Scholar

[2]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[3]

C. Bardos and K. D. Phung, Observation estimate for kinetic transport equations by diffusion approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 640-664.  doi: 10.1016/j.crma.2017.04.017.  Google Scholar

[4]

C. Bardos and L. Tartar, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal., 50 (1973), 10-25.  doi: 10.1007/BF00251291.  Google Scholar

[5]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., 7 (2002), 585-599.  doi: 10.1155/S108533750220408X.  Google Scholar

[6]

F. Chaves-Silva and G. Lebeau, Spectral inequality and optimal cost of controllability for the Stokes system, ESAIM Control Optim. Calc. Var., 22 (2016), 1137-1162.  doi: 10.1051/cocv/2016034.  Google Scholar

[7]

L. EscauriazaF. J. Fernandez and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.  doi: 10.1080/00036810500277082.  Google Scholar

[8]

L. EscauriazaC. KenigG. Ponce and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay, Math. Res. Lett., 15 (2008), 957-971.  doi: 10.4310/MRL.2008.v15.n5.a10.  Google Scholar

[9]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.  Google Scholar

[10]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy uncertainty principle, convexity and parabolic evolutions, Comm. Math. Phys., 346 (2016), 667-678.  doi: 10.1007/s00220-015-2500-z.  Google Scholar

[11]

L. EscauriazaS. 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.  Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[13]

P. Gao, The Lebeau-Robbiano inequality for the one-dimensional fourth order elliptic operator and its application, ESAIM Control Optim. Calc. Var., 22 (2016), 811-831.  doi: 10.1051/cocv/2015030.  Google Scholar

[14]

A. Grigor'yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 353-362.  doi: 10.1017/S0308210500028511.  Google Scholar

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.  Google Scholar

[16]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, (1999), 223-239.  Google Scholar

[17]

J. Le RousseauM. Léautaud and L. Robbiano, Controllability of a parabolic system with a diffuse interface, J. Eur. Math. Soc. (JEMS), 15 (2013), 1485-1574.  doi: 10.4171/JEMS/397.  Google Scholar

[18]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[19]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.  Google Scholar

[20]

J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal., 195 (2010), 953-990.  doi: 10.1007/s00205-009-0242-9.  Google Scholar

[21]

J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math., 183 (2011), 245-336.  doi: 10.1007/s00222-010-0278-3.  Google Scholar

[22]

J. Le Rousseau and L. Robbiano, Spectral inequality and resolvent estimate for the bi-Laplace operator, preprint, arXiv: 1509.02098. Google Scholar

[23]

M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., 258 (2010), 2739-2778.  doi: 10.1016/j.jfa.2009.10.011.  Google Scholar

[24]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[25]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.  doi: 10.1007/s002050050078.  Google Scholar

[26]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[27]

F. Lin, Remarks on a backward parabolic problem, Methods Appl. Anal., 10 (2003), 245-252.  doi: 10.4310/MAA.2003.v10.n2.a5.  Google Scholar

[28]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[29]

L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465.  Google Scholar

[30]

L. Payne, Improperly Posed Problems in Partial Differential Equations, Regional Conference Series in Applied Mathematics, Vol. 22, SIAM, 1975.  Google Scholar

[31]

K. D. Phung, Note on the cost of the approximate controllability for the heat equation with potential, J. Math. Anal. Appl., 295 (2004), 527-538.  doi: 10.1016/j.jmaa.2004.03.059.  Google Scholar

[32]

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.  Google Scholar

[33]

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. (JEMS), 15 (2013), 681-703.  doi: 10.4171/JEMS/371.  Google Scholar

[34]

K. D. PhungG. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, J. Differential Equations, 263 (2017), 5012-5041.  doi: 10.1016/j.jde.2017.06.008.  Google Scholar

[35]

K.D. PhungL. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 477-499.  doi: 10.1016/j.anihpc.2013.04.005.  Google Scholar

[36]

C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539.  doi: 10.1080/03605309608821195.  Google Scholar

[37]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[38]

S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.  Google Scholar

[39]

T. M. N. Vo, The local backward heat problem, preprint arXiv: 1704.05314. Google Scholar

[40]

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.  Google Scholar

[41]

X. Yu and L. Zhang, The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian, ESAIM: COCV. doi: 10.1051/cocv/2017075.  Google Scholar

[42]

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.  Google Scholar

show all references

References:
[1]

S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach Space, Comm. Pure Appl. Math., 16 (1963), 121-239.  doi: 10.1002/cpa.3160160204.  Google Scholar

[2]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[3]

C. Bardos and K. D. Phung, Observation estimate for kinetic transport equations by diffusion approximation, C. R. Math. Acad. Sci. Paris, 355 (2017), 640-664.  doi: 10.1016/j.crma.2017.04.017.  Google Scholar

[4]

C. Bardos and L. Tartar, Sur l'unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal., 50 (1973), 10-25.  doi: 10.1007/BF00251291.  Google Scholar

[5]

A. Benabdallah and M. G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal., 7 (2002), 585-599.  doi: 10.1155/S108533750220408X.  Google Scholar

[6]

F. Chaves-Silva and G. Lebeau, Spectral inequality and optimal cost of controllability for the Stokes system, ESAIM Control Optim. Calc. Var., 22 (2016), 1137-1162.  doi: 10.1051/cocv/2016034.  Google Scholar

[7]

L. EscauriazaF. J. Fernandez and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.  doi: 10.1080/00036810500277082.  Google Scholar

[8]

L. EscauriazaC. KenigG. Ponce and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay, Math. Res. Lett., 15 (2008), 957-971.  doi: 10.4310/MRL.2008.v15.n5.a10.  Google Scholar

[9]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.  Google Scholar

[10]

L. EscauriazaC. KenigG. Ponce and L. Vega, Hardy uncertainty principle, convexity and parabolic evolutions, Comm. Math. Phys., 346 (2016), 667-678.  doi: 10.1007/s00220-015-2500-z.  Google Scholar

[11]

L. EscauriazaS. 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.  Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[13]

P. Gao, The Lebeau-Robbiano inequality for the one-dimensional fourth order elliptic operator and its application, ESAIM Control Optim. Calc. Var., 22 (2016), 811-831.  doi: 10.1051/cocv/2015030.  Google Scholar

[14]

A. Grigor'yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 353-362.  doi: 10.1017/S0308210500028511.  Google Scholar

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.  Google Scholar

[16]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, (1999), 223-239.  Google Scholar

[17]

J. Le RousseauM. Léautaud and L. Robbiano, Controllability of a parabolic system with a diffuse interface, J. Eur. Math. Soc. (JEMS), 15 (2013), 1485-1574.  doi: 10.4171/JEMS/397.  Google Scholar

[18]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[19]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.  Google Scholar

[20]

J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Ration. Mech. Anal., 195 (2010), 953-990.  doi: 10.1007/s00205-009-0242-9.  Google Scholar

[21]

J. Le Rousseau and L. Robbiano, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math., 183 (2011), 245-336.  doi: 10.1007/s00222-010-0278-3.  Google Scholar

[22]

J. Le Rousseau and L. Robbiano, Spectral inequality and resolvent estimate for the bi-Laplace operator, preprint, arXiv: 1509.02098. Google Scholar

[23]

M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal., 258 (2010), 2739-2778.  doi: 10.1016/j.jfa.2009.10.011.  Google Scholar

[24]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[25]

G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), 297-329.  doi: 10.1007/s002050050078.  Google Scholar

[26]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[27]

F. Lin, Remarks on a backward parabolic problem, Methods Appl. Anal., 10 (2003), 245-252.  doi: 10.4310/MAA.2003.v10.n2.a5.  Google Scholar

[28]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[29]

L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465.  Google Scholar

[30]

L. Payne, Improperly Posed Problems in Partial Differential Equations, Regional Conference Series in Applied Mathematics, Vol. 22, SIAM, 1975.  Google Scholar

[31]

K. D. Phung, Note on the cost of the approximate controllability for the heat equation with potential, J. Math. Anal. Appl., 295 (2004), 527-538.  doi: 10.1016/j.jmaa.2004.03.059.  Google Scholar

[32]

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.  Google Scholar

[33]

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. (JEMS), 15 (2013), 681-703.  doi: 10.4171/JEMS/371.  Google Scholar

[34]

K. D. PhungG. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, J. Differential Equations, 263 (2017), 5012-5041.  doi: 10.1016/j.jde.2017.06.008.  Google Scholar

[35]

K.D. PhungL. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 477-499.  doi: 10.1016/j.anihpc.2013.04.005.  Google Scholar

[36]

C. C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539.  doi: 10.1080/03605309608821195.  Google Scholar

[37]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[38]

S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.  Google Scholar

[39]

T. M. N. Vo, The local backward heat problem, preprint arXiv: 1704.05314. Google Scholar

[40]

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.  Google Scholar

[41]

X. Yu and L. Zhang, The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian, ESAIM: COCV. doi: 10.1051/cocv/2017075.  Google Scholar

[42]

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.  Google Scholar

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