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

Abstract

Abstract. 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.

Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems

“…Such results can be very useful to address systems of PDEs, in particular for the purpose of solving inverse problems. On such questions see for instance [10,12,24,5].…”

Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases

“…Thanks to the second large parameter in our Carleman estimate (17) for a scalar hyperbolic equation, we will derive Carleman estimates for some strongly coupled systems. Isakov and Kim [17,18] apply Carleman estimates with second large parameter to a linear elastic system with residual stress, and as for Carleman estimates for other thermoelasticity systems, see Eller [19], Eller and Isakov [20]. In this section, thanks to Theorem 1, we establish Carleman estimates for .…”

“…However, in establishing the unique continuation and the observability inequality, and solving inverse problems for some systems in the mathematical physics such as the thermoelasticity system, we need a Carleman estimate with second large parameter , where the weight function ' is given in the form of e . Such Carleman estimates are proved in Isakov and Kim [17,18] and also see Eller [19], Eller and Isakov [20], where functions under consideration are assumed to have compact supports.…”

“…In this section we will use the scalar Carleman estimate with second large parameter (17) to prove Carleman estimates for a parabolic-hyperbolic strongly coupled system arising in the thermoelasticity theory. In order to formulate our Carleman type estimates we introduce some notations:…”

“…Indeed, in deriving a Carleman estimate for thermoelasticity systems, there is a difficulty coming from the coupling of the two equations and we have to keep the dependency on the second large parameter in the weight function. Thanks to the second large parameter in our Carleman estimate (17) for the scalar hyperbolic equation, we can derive Carleman estimates for some strongly coupled systems (Section 4).…”