American Institute of Mathematical Sciences

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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

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), 161-204.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  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), 973-978.  Google Scholar

[11]

P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715 (electronic).  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), 1-19.  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), 607-635.  Google Scholar

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.  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), 147-152.  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), 105003. 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), 831-878.  Google Scholar

[19]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.  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, Ellipses, Paris, 1990.  Google Scholar

[22]

E. B. Davies, "Spectral Theory and Differential Operators," Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 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), 1996-2019.  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), 139-148.  Google Scholar

[25]

E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM: Control, Optim, Calv. Var., 2 (1997), 87-103 (electronic).  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), 583-616.  Google Scholar

[27]

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

[28]

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

[29]

G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," 2nd ed., Cambridge, at the University Press, 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 "Partial Differential Equations Methods in Control and Shape Analysis," Lect. Notes in Pure and Applied Math., 188, Marcel Dekker, New York, (1994), 215-243.  Google Scholar

[31]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. in PDE, 20 (1995), 335-356.  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), 709-728.  Google Scholar

[33]

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

[34]

V. G. Maz'ja, "Sobolev Spaces," Springer-Verlag, Berlin, 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), 379-382.  Google Scholar

[36]

B. Opic and A. Kufner, "Hardy-Type Inequalities," Pitman Research Notes in Math., 219, Longman, 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), 367-408.  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, Ser. I, 348 (2010), 801-805. 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), 1864-1902.  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), 1508-1532.  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), 103-153.  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), 227-274.  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), 161-204.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  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), 973-978.  Google Scholar

[11]

P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715 (electronic).  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), 1-19.  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), 607-635.  Google Scholar

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.  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), 147-152.  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), 105003. 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), 831-878.  Google Scholar

[19]

M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.  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, Ellipses, Paris, 1990.  Google Scholar

[22]

E. B. Davies, "Spectral Theory and Differential Operators," Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 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), 1996-2019.  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), 139-148.  Google Scholar

[25]

E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM: Control, Optim, Calv. Var., 2 (1997), 87-103 (electronic).  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), 583-616.  Google Scholar

[27]

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

[28]

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

[29]

G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," 2nd ed., Cambridge, at the University Press, 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 "Partial Differential Equations Methods in Control and Shape Analysis," Lect. Notes in Pure and Applied Math., 188, Marcel Dekker, New York, (1994), 215-243.  Google Scholar

[31]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. in PDE, 20 (1995), 335-356.  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), 709-728.  Google Scholar

[33]

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

[34]

V. G. Maz'ja, "Sobolev Spaces," Springer-Verlag, Berlin, 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), 379-382.  Google Scholar

[36]

B. Opic and A. Kufner, "Hardy-Type Inequalities," Pitman Research Notes in Math., 219, Longman, 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), 367-408.  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, Ser. I, 348 (2010), 801-805. 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), 1864-1902.  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), 1508-1532.  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), 103-153.  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), 227-274.  Google Scholar

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