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December  2013, 2(4): 679-693. doi: 10.3934/eect.2013.2.679

Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress

Nanhee Kim 1,

1. 

Wichita State University, 1845 Fairmount, Wichita, KS 67260-0033, United States

Received  December 2012 Revised  July 2013 Published  November 2013

By introducing some auxiliary functions, an elasticity system with thermal effects becomes a coupled hyperbolic-parabolic system. Using this reduced system, we obtain a Carleman estimate with two large parameters for the linear thermoelasticity system with residual stress which is the basic tool for showing stability estimates in the lateral Cauchy problem.
Keywords: residual stress., Carleman estimate, Partial differential equations, Cauchy problem, thermoelasticity.
Mathematics Subject Classification: Primary: 35L51, 35M30; Secondary: 35K10, 58J3.
Citation: Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations & Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679
References:
[1]

P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system,, Electronic J. of Diff. Eqns., 22 (2000), 1.   Google Scholar

[2]

M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the themoelasticity system,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/1/015006.  Google Scholar

[3]

M. Eller and V. Isakov, Carleman estimates with two large parameters and applications,, Contemp. Math., 268 (2000), 117.  doi: 10.1090/conm/268/04310.  Google Scholar

[4]

M. Eller, I. Lasiecka and R. Triggiani, Simultaneous Exact/Approximate Boundary Controllability of Thermo-Elastic Plates with Variable Thermal Coefficient and Moment Control,, J. of Mathematical Analysis and Applications, 251 (2000), 452.  doi: 10.1006/jmaa.2000.7015.  Google Scholar

[5]

L. Hörmander, Linear Partial Differential Operators,, Springer-Verlag, (1976).   Google Scholar

[6]

V. Isakov, A Nonhyperbolic Cauchy Problem for $\square_b$, $square_c$ and its Applications to elasticity Theory,, Comm. Pure Appl. Math., 39 (1986), 747.  doi: 10.1002/cpa.3160390603.  Google Scholar

[7]

V. Isakov, On the uniqueness of the continuation for a thermoelasticity system,, SIAM J. Math. Anal., 33 (2001), 509.  doi: 10.1137/S0036141000366509.  Google Scholar

[8]

V. Isakov, Inverse Problems for Partial Differential Equations,, Second edition. Applied Mathematical Sciences, (2006).   Google Scholar

[9]

V. Isakov, Carleman estimates for some anisotropic elasticity systems and applications,, Evolution Equations and Control Theory, 1 (2012), 141.  doi: 10.3934/eect.2012.1.141.  Google Scholar

[10]

V. Isakov, G. Nakamura and J.-N. Wang, Uniqueness and stability in the Cauchy problem for the elasticity system with residual stress,, Contemp. Math. AMS, 333 (2003), 99.  doi: 10.1090/conm/333/05957.  Google Scholar

[11]

V. Isakov and N. Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress,, Applicationes Mathematicae, 35 (2008), 447.  doi: 10.4064/am35-4-4.  Google Scholar

[12]

V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators,, Some application of Sobolev spaces to PDEs, 10 (2009), 135.  doi: 10.1007/978-0-387-85652-0_3.  Google Scholar

[13]

V. Isakov and N. Kim, Weak Carleman estimates with large parameters for second order operators and applications to elasticity with residual stress,, Discrete Cont. Dyn. Systems-A, 27 (2010), 799.  doi: 10.3934/dcds.2010.27.799.  Google Scholar

[14]

I. Lasiecka and R. Triggiani, Analyticity of thermo-Elastic semigroups with free boundary conditions,, Ann. Scuola Norm. Sup. Pisa CI. Sci., 27 (1998), 457.   Google Scholar

[15]

C.-S. Man, Hartig's law and linear elasticity with initial stress,, Inverse Problems, 14 (1998), 313.  doi: 10.1088/0266-5611/14/2/007.  Google Scholar

[16]

B. Wu and J. Liu, Determination of an unknown source for a thermoelastic system with a memory effect,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/9/095012.  Google Scholar

show all references

References:
[1]

P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system,, Electronic J. of Diff. Eqns., 22 (2000), 1.   Google Scholar

[2]

M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the themoelasticity system,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/1/015006.  Google Scholar

[3]

M. Eller and V. Isakov, Carleman estimates with two large parameters and applications,, Contemp. Math., 268 (2000), 117.  doi: 10.1090/conm/268/04310.  Google Scholar

[4]

M. Eller, I. Lasiecka and R. Triggiani, Simultaneous Exact/Approximate Boundary Controllability of Thermo-Elastic Plates with Variable Thermal Coefficient and Moment Control,, J. of Mathematical Analysis and Applications, 251 (2000), 452.  doi: 10.1006/jmaa.2000.7015.  Google Scholar

[5]

L. Hörmander, Linear Partial Differential Operators,, Springer-Verlag, (1976).   Google Scholar

[6]

V. Isakov, A Nonhyperbolic Cauchy Problem for $\square_b$, $square_c$ and its Applications to elasticity Theory,, Comm. Pure Appl. Math., 39 (1986), 747.  doi: 10.1002/cpa.3160390603.  Google Scholar

[7]

V. Isakov, On the uniqueness of the continuation for a thermoelasticity system,, SIAM J. Math. Anal., 33 (2001), 509.  doi: 10.1137/S0036141000366509.  Google Scholar

[8]

V. Isakov, Inverse Problems for Partial Differential Equations,, Second edition. Applied Mathematical Sciences, (2006).   Google Scholar

[9]

V. Isakov, Carleman estimates for some anisotropic elasticity systems and applications,, Evolution Equations and Control Theory, 1 (2012), 141.  doi: 10.3934/eect.2012.1.141.  Google Scholar

[10]

V. Isakov, G. Nakamura and J.-N. Wang, Uniqueness and stability in the Cauchy problem for the elasticity system with residual stress,, Contemp. Math. AMS, 333 (2003), 99.  doi: 10.1090/conm/333/05957.  Google Scholar

[11]

V. Isakov and N. Kim, Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress,, Applicationes Mathematicae, 35 (2008), 447.  doi: 10.4064/am35-4-4.  Google Scholar

[12]

V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators,, Some application of Sobolev spaces to PDEs, 10 (2009), 135.  doi: 10.1007/978-0-387-85652-0_3.  Google Scholar

[13]

V. Isakov and N. Kim, Weak Carleman estimates with large parameters for second order operators and applications to elasticity with residual stress,, Discrete Cont. Dyn. Systems-A, 27 (2010), 799.  doi: 10.3934/dcds.2010.27.799.  Google Scholar

[14]

I. Lasiecka and R. Triggiani, Analyticity of thermo-Elastic semigroups with free boundary conditions,, Ann. Scuola Norm. Sup. Pisa CI. Sci., 27 (1998), 457.   Google Scholar

[15]

C.-S. Man, Hartig's law and linear elasticity with initial stress,, Inverse Problems, 14 (1998), 313.  doi: 10.1088/0266-5611/14/2/007.  Google Scholar

[16]

B. Wu and J. Liu, Determination of an unknown source for a thermoelastic system with a memory effect,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/9/095012.  Google Scholar

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