December  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).   Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[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.   Google Scholar

[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.   Google Scholar

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

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).   Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[23]

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

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[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.   Google Scholar

[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.   Google Scholar

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

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