Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems

Judith Vancostenoble

doi:10.3934/dcdss.2011.4.761

Previous Article
DCDS-S Home
This Issue
Next Article
Riesz systems, spectral controllability and a source identification problem for heat equations with memory

June 2011, 4(3): 761-790. doi: 10.3934/dcdss.2011.4.761

Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems

Judith Vancostenoble ^1,

1.	Université de Toulouse; Université Paul Sabatier Toulouse III, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse, France

Received April 2009 Revised December 2009 Published November 2010

Abstract
Related Papers

We consider the following class of degenerate/singular parabolic operators:

$Pu=u_t-(x^\a u_x)_x-$λ$ u$/($x^$β) , $x\in (0,1)$,

associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters $\a\geq 0$, β, λ$ \in \mathbb R$, we derive sharp global Carleman estimates. As an application, we deduce observability and null controllability results for the corresponding evolution problem. A key step in the proof of Carleman estimates is the correct choice of the weight functions and a key ingredient in the proof takes the form of special Hardy-Poincaré inequalities

Keywords: degenerate diffusion, Hardy-Poincaré's inequalities, Carleman estimates, Parabolic equation, null controllability, singular potential, observability.

Mathematics Subject Classification: 93B05, 93C20, 93B07, 35K6.

Citation: Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761

References:

[1]	F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,, J. Evol. Equ., 6 (2006), 161. Google Scholar
[2]	S. Aniţa and D. Tataru, Null controllability for the dissipative semilinear heat equation,, Appl. Math. Optim., 46 (2002), 97. Google Scholar
[3]	B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion,, Abstr. Appl. Anal., 6 (2001), 357. Google Scholar
[4]	P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics,, in, 92 (1984), 31. Google Scholar
[5]	P. Baras and J. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. Google Scholar
[6]	K. Beauchard and E. Zuazua, Some controllability results for the 2D Kolmogorov equation,, Ann. Institut Henri Poincaré, 26 (2009), 1793. Google Scholar
[7]	J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Math. Sci., 83 (1989). Google Scholar
[8]	H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Complut., 10 (1997), 443. Google Scholar
[9]	J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations,, in, 139 (2002), 31. Google Scholar
[10]	X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973. Google Scholar
[11]	P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability of degenerate parabolic operators with drift,, Netw. Heterog. Media, 2 (2007), 695. Google Scholar
[12]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM J. Control Optim., 47 (2008), 1. Google Scholar
[13]	P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Commun. Pure Appl. Anal., 3 (2004), 607. Google Scholar
[14]	P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations,, Adv. Differential Equations, 10 (2005), 153. Google Scholar
[15]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators,, C. R. Acad. Sci. Paris Sér. I Math., 347 (2009), 147. Google Scholar
[16]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications,, AMS Memoirs, (). Google Scholar
[17]	P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation,, Inverse Problems, 26 (2010). Google Scholar
[18]	P. Cannarsa, D. Rocchetti and J. Vancostenoble, Generation of analytic semi-groups in $L^2$ for a class of second order degenerate elliptic operators,, Control Cybernet., 37 (2008), 831. Google Scholar
[19]	M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval,, Semigroup Forum, 57 (1998), 1. Google Scholar
[20]	S. Chandrasekhar, "An Introduction to the Study of Stellar Structure,", Dover Publ. Inc. New York, (1985). Google Scholar
[21]	T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires,, Mathématiques et Applications, (1990). Google Scholar
[22]	E. B. Davies, "Spectral Theory and Differential Operators,", Cambridge Studies in Advanced Mathematics, 42 (1995). Google Scholar
[23]	S. Ervedoza, Null Controllability for a singular heat equation: Carleman estimates and Hardy inequalities,, Com. in Partial Diff. Eq., 33 (2008), 1996. Google Scholar
[24]	L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space,, St. Petersburg Math. J., 15 (2004), 139. Google Scholar
[25]	E. Fernández-Cara, Null controllability of the semilinear heat equation,, ESAIM: Control, 2 (1997), 87. Google Scholar
[26]	E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583. Google Scholar
[27]	A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Series, 34 (1996). Google Scholar
[28]	J. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. Amer. Math. Soc., 355 (2003), 197. Google Scholar
[29]	G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", 2nd ed., (1952). Google Scholar
[30]	I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, non conservative second order hyperbolic equations,, in, 188 (1994), 215. Google Scholar
[31]	G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. in PDE, 20 (1995), 335. Google Scholar
[32]	P. Martinez, J.-P. Raymond and J. Vancostenoble, Regional null controllability of a Crocco type linearized equation,, SIAM J. Control Optim., 42 (2003), 709. Google Scholar
[33]	P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations,, J. Evol. Equ., 6 (2006), 325. Google Scholar
[34]	V. G. Maz'ja, "Sobolev Spaces,", Springer-Verlag, (1985). Google Scholar
[35]	F. Mignot and J.-P. Puel, Solution radiale singulière de $-\Delta u=\lambda e^u$,, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 379. Google Scholar
[36]	B. Opic and A. Kufner, "Hardy-Type Inequalities,", Pitman Research Notes in Math., 219 (1990). Google Scholar
[37]	D. Tataru, Carleman estimates and unique continuation near the boundary for P.D.E.'s,, Journal de Maths. Pures et Appliquées, 75 (1996), 367. Google Scholar
[38]	J. Vancostenoble, Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems,, C. R. Acad. Sci. Paris, 348 (2010), 801. Google Scholar
[39]	J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations,, Comm. in PDE, (). Google Scholar
[40]	J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials,, J. Funct. Anal., 254 (2008), 1864. Google Scholar
[41]	J. Vancostenoble and E. Zuazua, Hardy inequalities, Observability and Control for the wave and Schrödinger equations with singular potentials,, SIAM J. Math. Anal., 41 (2009), 1508. Google Scholar
[42]	J. L. Vázquez 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. Google Scholar
[43]	O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations,, Publ. Res. Inst. Math. Sci., 39 (2003), 227. Google Scholar

show all references

References:

[1]	F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,, J. Evol. Equ., 6 (2006), 161. Google Scholar
[2]	S. Aniţa and D. Tataru, Null controllability for the dissipative semilinear heat equation,, Appl. Math. Optim., 46 (2002), 97. Google Scholar
[3]	B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion,, Abstr. Appl. Anal., 6 (2001), 357. Google Scholar
[4]	P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics,, in, 92 (1984), 31. Google Scholar
[5]	P. Baras and J. Goldstein, The heat equation with a singular potential,, Trans. Amer. Math. Soc., 284 (1984), 121. Google Scholar
[6]	K. Beauchard and E. Zuazua, Some controllability results for the 2D Kolmogorov equation,, Ann. Institut Henri Poincaré, 26 (2009), 1793. Google Scholar
[7]	J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Math. Sci., 83 (1989). Google Scholar
[8]	H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,, Rev. Mat. Complut., 10 (1997), 443. Google Scholar
[9]	J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations,, in, 139 (2002), 31. Google Scholar
[10]	X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973. Google Scholar
[11]	P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability of degenerate parabolic operators with drift,, Netw. Heterog. Media, 2 (2007), 695. Google Scholar
[12]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM J. Control Optim., 47 (2008), 1. Google Scholar
[13]	P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Commun. Pure Appl. Anal., 3 (2004), 607. Google Scholar
[14]	P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations,, Adv. Differential Equations, 10 (2005), 153. Google Scholar
[15]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates and null controllability for boundary-degenerate parabolic operators,, C. R. Acad. Sci. Paris Sér. I Math., 347 (2009), 147. Google Scholar
[16]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications,, AMS Memoirs, (). Google Scholar
[17]	P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation,, Inverse Problems, 26 (2010). Google Scholar
[18]	P. Cannarsa, D. Rocchetti and J. Vancostenoble, Generation of analytic semi-groups in $L^2$ for a class of second order degenerate elliptic operators,, Control Cybernet., 37 (2008), 831. Google Scholar
[19]	M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval,, Semigroup Forum, 57 (1998), 1. Google Scholar
[20]	S. Chandrasekhar, "An Introduction to the Study of Stellar Structure,", Dover Publ. Inc. New York, (1985). Google Scholar
[21]	T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires,, Mathématiques et Applications, (1990). Google Scholar
[22]	E. B. Davies, "Spectral Theory and Differential Operators,", Cambridge Studies in Advanced Mathematics, 42 (1995). Google Scholar
[23]	S. Ervedoza, Null Controllability for a singular heat equation: Carleman estimates and Hardy inequalities,, Com. in Partial Diff. Eq., 33 (2008), 1996. Google Scholar
[24]	L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space,, St. Petersburg Math. J., 15 (2004), 139. Google Scholar
[25]	E. Fernández-Cara, Null controllability of the semilinear heat equation,, ESAIM: Control, 2 (1997), 87. Google Scholar
[26]	E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583. Google Scholar
[27]	A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Series, 34 (1996). Google Scholar
[28]	J. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. Amer. Math. Soc., 355 (2003), 197. Google Scholar
[29]	G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", 2nd ed., (1952). Google Scholar
[30]	I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, non conservative second order hyperbolic equations,, in, 188 (1994), 215. Google Scholar
[31]	G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. in PDE, 20 (1995), 335. Google Scholar
[32]	P. Martinez, J.-P. Raymond and J. Vancostenoble, Regional null controllability of a Crocco type linearized equation,, SIAM J. Control Optim., 42 (2003), 709. Google Scholar
[33]	P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations,, J. Evol. Equ., 6 (2006), 325. Google Scholar
[34]	V. G. Maz'ja, "Sobolev Spaces,", Springer-Verlag, (1985). Google Scholar
[35]	F. Mignot and J.-P. Puel, Solution radiale singulière de $-\Delta u=\lambda e^u$,, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 379. Google Scholar
[36]	B. Opic and A. Kufner, "Hardy-Type Inequalities,", Pitman Research Notes in Math., 219 (1990). Google Scholar
[37]	D. Tataru, Carleman estimates and unique continuation near the boundary for P.D.E.'s,, Journal de Maths. Pures et Appliquées, 75 (1996), 367. Google Scholar
[38]	J. Vancostenoble, Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems,, C. R. Acad. Sci. Paris, 348 (2010), 801. Google Scholar
[39]	J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations,, Comm. in PDE, (). Google Scholar
[40]	J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials,, J. Funct. Anal., 254 (2008), 1864. Google Scholar
[41]	J. Vancostenoble and E. Zuazua, Hardy inequalities, Observability and Control for the wave and Schrödinger equations with singular potentials,, SIAM J. Math. Anal., 41 (2009), 1508. Google Scholar
[42]	J. L. Vázquez 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. Google Scholar
[43]	O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations,, Publ. Res. Inst. Math. Sci., 39 (2003), 227. Google Scholar

[1]	El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441
[2]	Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687
[3]	Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695
[4]	Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control & Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203
[5]	Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020080
[6]	J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020136
[7]	Abdelhakim Belghazi, Ferroudja Smadhi, Nawel Zaidi, Ouahiba Zair. Carleman inequalities for the two-dimensional heat equation in singular domains. Mathematical Control & Related Fields, 2012, 2 (4) : 331-359. doi: 10.3934/mcrf.2012.2.331
[8]	Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133
[9]	Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1
[10]	Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415
[11]	Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[12]	Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037
[13]	El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control & Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013
[14]	Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[15]	Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031
[16]	Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[17]	Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001
[18]	Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[19]	Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116
[20]	Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control & Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004

2019 Impact Factor: 1.233

American Institute of Mathematical Sciences

Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors