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

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

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

Citation: Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete and 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), 161-204.

[2]

S. Aniţa and D. Tataru, Null controllability for the dissipative semilinear heat equation, Appl. Math. Optim., 46 (2002), 97-105.

[3]

B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal., 6 (2001), 357-368.

[4]

P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics, in "Differential Equations" (Birmingham, Ala., 1983), North-Holland Math. Stud., 92, North-Holland, Amsterdam, (1984), 31-35.

[5]

P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.

[6]

K. Beauchard and E. Zuazua, Some controllability results for the 2D Kolmogorov equation, Ann. Institut Henri Poincaré, Analyse Non Linéaire, 26 (2009), 1793-1815.

[7]

J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory," Math. Sci., Vol. 83, Springer-Verlag, New York, 1989.

[8]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Complut., 10 (1997), 443-469.

[9]

J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations, in "Optimal Control of Complex Structures" (Oberwolfach, 2000), Internat. Ser. Numer. Math., 139, Birkhauser, Basel, (2002) 31-42.

[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), 973-978.

[11]

P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715 (electronic).

[12]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.

[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), 607-635.

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.

[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), 147-152.

[16]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications,, AMS Memoirs, (). 

[17]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003.

[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), 831-878.

[19]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.

[20]

S. Chandrasekhar, "An Introduction to the Study of Stellar Structure," Dover Publ. Inc. New York, 1985.

[21]

T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Mathématiques et Applications, Ellipses, Paris, 1990.

[22]

E. B. Davies, "Spectral Theory and Differential Operators," Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995.

[23]

S. Ervedoza, Null Controllability for a singular heat equation: Carleman estimates and Hardy inequalities, Com. in Partial Diff. Eq., 33 (2008), 1996-2019.

[24]

L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, St. Petersburg Math. J., 15 (2004), 139-148.

[25]

E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM: Control, Optim, Calv. Var., 2 (1997), 87-103 (electronic).

[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), 583-616.

[27]

A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University, Seoul, Korea, 1996.

[28]

J. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc., 355 (2003), 197-211 (electronic).

[29]

G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," 2nd ed., Cambridge, at the University Press, 1952.

[30]

I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, non conservative second order hyperbolic equations, in "Partial Differential Equations Methods in Control and Shape Analysis," Lect. Notes in Pure and Applied Math., 188, Marcel Dekker, New York, (1994), 215-243.

[31]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. in PDE, 20 (1995), 335-356.

[32]

P. Martinez, J.-P. Raymond and J. Vancostenoble, Regional null controllability of a Crocco type linearized equation, SIAM J. Control Optim., 42 (2003), 709-728.

[33]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.

[34]

V. G. Maz'ja, "Sobolev Spaces," Springer-Verlag, Berlin, 1985.

[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), 379-382.

[36]

B. Opic and A. Kufner, "Hardy-Type Inequalities," Pitman Research Notes in Math., 219, Longman, 1990.

[37]

D. Tataru, Carleman estimates and unique continuation near the boundary for P.D.E.'s, Journal de Maths. Pures et Appliquées, 75 (1996), 367-408.

[38]

J. Vancostenoble, Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems, C. R. Acad. Sci. Paris, Ser. I, 348 (2010), 801-805.

[39]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations,, Comm. in PDE, (). 

[40]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.

[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), 1508-1532.

[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), 103-153.

[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), 227-274.

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), 161-204.

[2]

S. Aniţa and D. Tataru, Null controllability for the dissipative semilinear heat equation, Appl. Math. Optim., 46 (2002), 97-105.

[3]

B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal., 6 (2001), 357-368.

[4]

P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics, in "Differential Equations" (Birmingham, Ala., 1983), North-Holland Math. Stud., 92, North-Holland, Amsterdam, (1984), 31-35.

[5]

P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.

[6]

K. Beauchard and E. Zuazua, Some controllability results for the 2D Kolmogorov equation, Ann. Institut Henri Poincaré, Analyse Non Linéaire, 26 (2009), 1793-1815.

[7]

J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory," Math. Sci., Vol. 83, Springer-Verlag, New York, 1989.

[8]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Complut., 10 (1997), 443-469.

[9]

J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations, in "Optimal Control of Complex Structures" (Oberwolfach, 2000), Internat. Ser. Numer. Math., 139, Birkhauser, Basel, (2002) 31-42.

[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), 973-978.

[11]

P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715 (electronic).

[12]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.

[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), 607-635.

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.

[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), 147-152.

[16]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications,, AMS Memoirs, (). 

[17]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003.

[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), 831-878.

[19]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.

[20]

S. Chandrasekhar, "An Introduction to the Study of Stellar Structure," Dover Publ. Inc. New York, 1985.

[21]

T. Cazenave and A. Haraux, Introduction aux problèmes d'évolution semi-linéaires, Mathématiques et Applications, Ellipses, Paris, 1990.

[22]

E. B. Davies, "Spectral Theory and Differential Operators," Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995.

[23]

S. Ervedoza, Null Controllability for a singular heat equation: Carleman estimates and Hardy inequalities, Com. in Partial Diff. Eq., 33 (2008), 1996-2019.

[24]

L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, St. Petersburg Math. J., 15 (2004), 139-148.

[25]

E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM: Control, Optim, Calv. Var., 2 (1997), 87-103 (electronic).

[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), 583-616.

[27]

A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series, 34, Seoul National University, Seoul, Korea, 1996.

[28]

J. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc., 355 (2003), 197-211 (electronic).

[29]

G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," 2nd ed., Cambridge, at the University Press, 1952.

[30]

I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, non conservative second order hyperbolic equations, in "Partial Differential Equations Methods in Control and Shape Analysis," Lect. Notes in Pure and Applied Math., 188, Marcel Dekker, New York, (1994), 215-243.

[31]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. in PDE, 20 (1995), 335-356.

[32]

P. Martinez, J.-P. Raymond and J. Vancostenoble, Regional null controllability of a Crocco type linearized equation, SIAM J. Control Optim., 42 (2003), 709-728.

[33]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.

[34]

V. G. Maz'ja, "Sobolev Spaces," Springer-Verlag, Berlin, 1985.

[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), 379-382.

[36]

B. Opic and A. Kufner, "Hardy-Type Inequalities," Pitman Research Notes in Math., 219, Longman, 1990.

[37]

D. Tataru, Carleman estimates and unique continuation near the boundary for P.D.E.'s, Journal de Maths. Pures et Appliquées, 75 (1996), 367-408.

[38]

J. Vancostenoble, Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems, C. R. Acad. Sci. Paris, Ser. I, 348 (2010), 801-805.

[39]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations,, Comm. in PDE, (). 

[40]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.

[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), 1508-1532.

[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), 103-153.

[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), 227-274.

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