Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress

Victor Isakov¹ ; Nanhee Kim¹

¹ Department of Mathematics and Statistics Wichita State University Wichita, KS 67206, U.S.A.

Applicationes Mathematicae, Tome 35 (2008) no. 4, pp. 447-465.

Résumé

We derive Carleman type estimates with two large parameters for a general partial differential operator of second order. The weight function is assumed to be pseudo-convex with respect to the operator. We give applications to uniqueness and stability of the continuation of solutions and identification of coefficients for the Lamé system of dynamical elasticity with residual stress. This system is anisotropic and cannot be principally diagonalized, but it can be transformed into an “upper triangular” form. The use of two large parameters is essential for obtaining our results without smallness assumptions on the residual stress. In the proofs we use the classical technique of differential quadratic forms combined with a special partitioning of these forms and demonstrating positivity of terms containing highest powers of the second large parameter.

Détail

DOI : 10.4064/am35-4-4

Keywords: derive carleman type estimates large parameters general partial differential operator second order weight function assumed pseudo convex respect operator applications uniqueness stability continuation solutions identification coefficients lam system dynamical elasticity residual stress system anisotropic cannot principally diagonalized transformed upper triangular form large parameters essential obtaining results without smallness assumptions residual stress proofs classical technique differential quadratic forms combined special partitioning these forms demonstrating positivity terms containing highest powers second large parameter