2012, 1(2): 271-296. doi: 10.3934/eect.2012.1.271

Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems

1. 

Department of Mathematics, Georgetown University, Washington, DC 20057, United States

2. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, United States

Received  July 2012 Revised  September 2012 Published  October 2012

A Carleman estimate for some first-order elliptic systems is established. This estimate is extended to elliptic boundary value problems provided the boundary condition satisfies a Lopatinskii-type requirement. Based on these estimates conservative hyperbolic systems of first order can be stabilized with a logarithmic decay rate by introducing a localized interior dissipation. The support of the dissipative term does not need to satisfy a geometric condition.
Citation: Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
References:
[1]

G. Avalos and R. Triggiani, Rational decay rates for a pde fluid-structure interaction: a frequency domain approach,, Preprint, (2012).

[2]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.

[3]

M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping,, J. Differential Equations, 211 (2005), 303.

[4]

S. Benzoni-Gavage and D. Serre, "Multidimensional Hyperbolic Partial Differential Equations,", Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, (2007).

[5]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0.

[6]

A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations,, Amer. J. Math., 80 (1958), 16. doi: 10.2307/2372819.

[7]

T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes,, Ark. Mat., 26 (1939).

[8]

D. Chae, O. Y. Imanuvilov and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions,, J. Dynam. Control Systems, 2 (1996), 449. doi: 10.1007/BF02254698.

[9]

J.-F. Coulombel, Well-posedness of hyperbolic initial boundary value problems,, J. Math. Pures Appl. (9), 84 (2005), 786.

[10]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Springer-Verlag, (1990).

[11]

M. Eller, On symmetric hyperbolic boundary problems with nonhomogeneous conservative boundary conditions,, SIAM Journal of Mathematical Analysis, 4 (2012), 1925. doi: 10.1137/110834652.

[12]

M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems,, in, XIV (2002), 329.

[13]

M. Eller and D. Toundykov, A global holmgren theorem for multidimensional hyperbolic partial differential equations,, Applicable Analysis, 91 (2012), 69. doi: 10.1080/00036811.2010.538685.

[14]

K. O. Friedrichs, Symmetric hyperbolic linear differential equations,, Comm. Pure Appl. Math., 7 (1954), 345. doi: 10.1002/cpa.3160070206.

[15]

L. Hörmander, "Linear Partial Differential Operators,", Die Grundlehren der mathematischen Wissenschaften, (1963).

[16]

L. Hörmander, "The Analysis of Linear Partial Differential Operators III,", vol. 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, (1994).

[17]

O. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data,, Comm. Pure Appl. Math., 56 (2003), 1366. doi: 10.1002/cpa.10097.

[18]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a stationary isotropic Lamé system and the applications,, Appl. Anal., 83 (2004), 243. doi: 10.1080/00036810310001632772.

[19]

V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217.

[20]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Springer-Verlag, (2006).

[21]

V. Isakov and N. Kim, Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress,, Discrete Contin. Dyn. Syst., 27 (2010), 799. doi: 10.3934/dcds.2010.27.799.

[22]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1976).

[23]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277. doi: 10.1002/cpa.3160230304.

[24]

E. M. Landis and O. A. Ole{\u\i}nik, Generalized analyticity and certain properties, of solutions of elliptic and parabolic equations, that are connected with it,, Uspehi Mat. Nauk, 29 (1974), 190.

[25]

P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators,, Comm. Pure Appl. Math., 13 (1960), 427. doi: 10.1002/cpa.3160130307.

[26]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335.

[27]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4.

[28]

G. Métivier, Stability of multidimensional shocks,, In, (2001), 25.

[29]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation,, SIAM J. Control Optim., 41 (2002), 1554.

[30]

L. Nirenberg., "Lectures on Linear Partial Differential Equations,", American Mathematical Society, (1973), 22.

[31]

T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary,, Hokkaido Math. J., 10 (1981), 93.

[32]

J. B. Rauch and F. J. Massey, III, Differentiability of solutions to hyperbolic initial-boundary value problems,, Trans. Amer. Math. Soc., 189 (1974), 303.

[33]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains,, Indiana Univ. Math. J. 24 (1974), 24 (1974), 79.

[34]

L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques,, Comm. Partial Differential Equations, 16 (1991), 789.

[35]

L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques,, Asymptotic Anal., 10 (1995), 95.

[36]

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl. (9), 75 (1996), 367.

[37]

M. E. Taylor, "Partial Differential Equations I. Basic theory,", vol. 115 of Applied Mathematical Sciences. Springer, (2011).

[38]

N. Weck, Auβenraumaufgaben in der Theorie stationärer Schwingungen inhomogener elastischer Körper,, Math. Z., 111 (1969), 387. doi: 10.1007/BF01110749.

show all references

References:
[1]

G. Avalos and R. Triggiani, Rational decay rates for a pde fluid-structure interaction: a frequency domain approach,, Preprint, (2012).

[2]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.

[3]

M. Bellassoued, Decay of solutions of the wave equation with arbitrary localized nonlinear damping,, J. Differential Equations, 211 (2005), 303.

[4]

S. Benzoni-Gavage and D. Serre, "Multidimensional Hyperbolic Partial Differential Equations,", Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, (2007).

[5]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0.

[6]

A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations,, Amer. J. Math., 80 (1958), 16. doi: 10.2307/2372819.

[7]

T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes,, Ark. Mat., 26 (1939).

[8]

D. Chae, O. Y. Imanuvilov and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions,, J. Dynam. Control Systems, 2 (1996), 449. doi: 10.1007/BF02254698.

[9]

J.-F. Coulombel, Well-posedness of hyperbolic initial boundary value problems,, J. Math. Pures Appl. (9), 84 (2005), 786.

[10]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,", Springer-Verlag, (1990).

[11]

M. Eller, On symmetric hyperbolic boundary problems with nonhomogeneous conservative boundary conditions,, SIAM Journal of Mathematical Analysis, 4 (2012), 1925. doi: 10.1137/110834652.

[12]

M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems,, in, XIV (2002), 329.

[13]

M. Eller and D. Toundykov, A global holmgren theorem for multidimensional hyperbolic partial differential equations,, Applicable Analysis, 91 (2012), 69. doi: 10.1080/00036811.2010.538685.

[14]

K. O. Friedrichs, Symmetric hyperbolic linear differential equations,, Comm. Pure Appl. Math., 7 (1954), 345. doi: 10.1002/cpa.3160070206.

[15]

L. Hörmander, "Linear Partial Differential Operators,", Die Grundlehren der mathematischen Wissenschaften, (1963).

[16]

L. Hörmander, "The Analysis of Linear Partial Differential Operators III,", vol. 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, (1994).

[17]

O. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data,, Comm. Pure Appl. Math., 56 (2003), 1366. doi: 10.1002/cpa.10097.

[18]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a stationary isotropic Lamé system and the applications,, Appl. Anal., 83 (2004), 243. doi: 10.1080/00036810310001632772.

[19]

V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217.

[20]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Springer-Verlag, (2006).

[21]

V. Isakov and N. Kim, Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress,, Discrete Contin. Dyn. Syst., 27 (2010), 799. doi: 10.3934/dcds.2010.27.799.

[22]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1976).

[23]

H.-O. Kreiss, Initial boundary value problems for hyperbolic systems,, Comm. Pure Appl. Math., 23 (1970), 277. doi: 10.1002/cpa.3160230304.

[24]

E. M. Landis and O. A. Ole{\u\i}nik, Generalized analyticity and certain properties, of solutions of elliptic and parabolic equations, that are connected with it,, Uspehi Mat. Nauk, 29 (1974), 190.

[25]

P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators,, Comm. Pure Appl. Math., 13 (1960), 427. doi: 10.1002/cpa.3160130307.

[26]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335.

[27]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4.

[28]

G. Métivier, Stability of multidimensional shocks,, In, (2001), 25.

[29]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation,, SIAM J. Control Optim., 41 (2002), 1554.

[30]

L. Nirenberg., "Lectures on Linear Partial Differential Equations,", American Mathematical Society, (1973), 22.

[31]

T. Ohkubo, Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary,, Hokkaido Math. J., 10 (1981), 93.

[32]

J. B. Rauch and F. J. Massey, III, Differentiability of solutions to hyperbolic initial-boundary value problems,, Trans. Amer. Math. Soc., 189 (1974), 303.

[33]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains,, Indiana Univ. Math. J. 24 (1974), 24 (1974), 79.

[34]

L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques,, Comm. Partial Differential Equations, 16 (1991), 789.

[35]

L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques,, Asymptotic Anal., 10 (1995), 95.

[36]

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl. (9), 75 (1996), 367.

[37]

M. E. Taylor, "Partial Differential Equations I. Basic theory,", vol. 115 of Applied Mathematical Sciences. Springer, (2011).

[38]

N. Weck, Auβenraumaufgaben in der Theorie stationärer Schwingungen inhomogener elastischer Körper,, Math. Z., 111 (1969), 387. doi: 10.1007/BF01110749.

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