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

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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

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

Received December 2012 Revised July 2013 Published November 2013

Abstract
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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 and 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), 1-15.
[2]	M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the themoelasticity system, Inverse Problems, 27 (2011), 18pp. doi: 10.1088/0266-5611/27/1/015006.
[3]	M. Eller and V. Isakov, Carleman estimates with two large parameters and applications, Contemp. Math., AMS, 268 (2000), 117-137. doi: 10.1090/conm/268/04310.
[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), 452-478. doi: 10.1006/jmaa.2000.7015.
[5]	L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin-New York, 1976.
[6]	V. Isakov, A Nonhyperbolic Cauchy Problem for $\square_b$, $square_c$ and its Applications to elasticity Theory, Comm. Pure Appl. Math., 39 (1986), 747-767. doi: 10.1002/cpa.3160390603.
[7]	V. Isakov, On the uniqueness of the continuation for a thermoelasticity system, SIAM J. Math. Anal., 33 (2001), 509-522. doi: 10.1137/S0036141000366509.
[8]	V. Isakov, Inverse Problems for Partial Differential Equations, Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006.
[9]	V. Isakov, Carleman estimates for some anisotropic elasticity systems and applications, Evolution Equations and Control Theory, 1 (2012), 141-154. doi: 10.3934/eect.2012.1.141.
[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), 99-113. doi: 10.1090/conm/333/05957.
[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), 447-465. doi: 10.4064/am35-4-4.
[12]	V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators, Some application of Sobolev spaces to PDEs, International Math. Ser., Springer-Verlag, 10 (2009), 135-159. doi: 10.1007/978-0-387-85652-0_3.
[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), 799-825. doi: 10.3934/dcds.2010.27.799.
[14]	I. Lasiecka and R. Triggiani, Analyticity of thermo-Elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa CI. Sci., 27 (1998), 457-482.
[15]	C.-S. Man, Hartig's law and linear elasticity with initial stress, Inverse Problems, 14 (1998), 313-319. doi: 10.1088/0266-5611/14/2/007.
[16]	B. Wu and J. Liu, Determination of an unknown source for a thermoelastic system with a memory effect, Inverse Problems, 28 (2012), 17pp. doi: 10.1088/0266-5611/28/9/095012.

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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), 1-15.
[2]	M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the themoelasticity system, Inverse Problems, 27 (2011), 18pp. doi: 10.1088/0266-5611/27/1/015006.
[3]	M. Eller and V. Isakov, Carleman estimates with two large parameters and applications, Contemp. Math., AMS, 268 (2000), 117-137. doi: 10.1090/conm/268/04310.
[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), 452-478. doi: 10.1006/jmaa.2000.7015.
[5]	L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin-New York, 1976.
[6]	V. Isakov, A Nonhyperbolic Cauchy Problem for $\square_b$, $square_c$ and its Applications to elasticity Theory, Comm. Pure Appl. Math., 39 (1986), 747-767. doi: 10.1002/cpa.3160390603.
[7]	V. Isakov, On the uniqueness of the continuation for a thermoelasticity system, SIAM J. Math. Anal., 33 (2001), 509-522. doi: 10.1137/S0036141000366509.
[8]	V. Isakov, Inverse Problems for Partial Differential Equations, Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006.
[9]	V. Isakov, Carleman estimates for some anisotropic elasticity systems and applications, Evolution Equations and Control Theory, 1 (2012), 141-154. doi: 10.3934/eect.2012.1.141.
[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), 99-113. doi: 10.1090/conm/333/05957.
[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), 447-465. doi: 10.4064/am35-4-4.
[12]	V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators, Some application of Sobolev spaces to PDEs, International Math. Ser., Springer-Verlag, 10 (2009), 135-159. doi: 10.1007/978-0-387-85652-0_3.
[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), 799-825. doi: 10.3934/dcds.2010.27.799.
[14]	I. Lasiecka and R. Triggiani, Analyticity of thermo-Elastic semigroups with free boundary conditions, Ann. Scuola Norm. Sup. Pisa CI. Sci., 27 (1998), 457-482.
[15]	C.-S. Man, Hartig's law and linear elasticity with initial stress, Inverse Problems, 14 (1998), 313-319. doi: 10.1088/0266-5611/14/2/007.
[16]	B. Wu and J. Liu, Determination of an unknown source for a thermoelastic system with a memory effect, Inverse Problems, 28 (2012), 17pp. doi: 10.1088/0266-5611/28/9/095012.

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