CARLEMAN ESTIMATES FOR SOME FIRST-ORDER SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS Matthias Eller Department of Mathematics, Georgetown University, Washington, DC 20057, USA, mme4@georgetown.edu Keywords: systems of partial differential equations, Carleman estimates 1. INTRODUCTION Carleman estimates were developed in order to prove uniqueness for non-hyperbolic Cauchy problems for operators with non-analytic coefficients. More recently they have become powerful tools for many problems in the control of partial differential equations. The theory for scalar equations is rather complete, however concerning systems of equations the picture is less clear. The only general result pertaining Carleman estimates for systems of equations is due to Calderón (1). In this talk we will establish Carleman estimates for certain first order systems. In contrast to previous works (see for example (2),(3) and the references therein) our result does not rely on pseudo-differential operators or diagonalization methods which allows for minimal smoothness assumptions on the coefficients. This is of some significance when considering nonlinear problems. Moreover, the explicit nature of the estimate makes the inclusion of boundary terms possible. 2. THE RESULT Consider the 4 × 4 matrix partial differential operator A(x, ∂)u = (∇ × u1 + ∇u2 , −∇ · u1 ) of first order where u = (u1 , u2 ) and u1 a vectorvalued function with three components and u2 a scalar-valued function. Let Ω ⊂ R3 be an open set and assume that ψ ∈ C 2 (Ω) with ∇ψ 6= 0 in Ω. Set φ = esψ − 1 where s ≥ s0 . By elementary methods we will prove the following Carleman estimate. There exist constants τ0 and C such that for τ ≥ τ0 Z Z τ e2τ φ |u|2 dx ≤ C e2τ φ |A(x, ∂)u|2 dx Ω Ω for all compactly supported functions u ∈ C0∞ (Ω). We will also discuss the case of variable coefficients Aα (x, ∂)u = (∇ × u1 + α∇u2 , −∇ · (αu1 )) where α ∈ C 1 (Ω) and the dynamic case, i.e. the operator   ∂t − A(x, ∂) P (x, ∂) = ∂t + A(x, ∂) acting on a vector-valued function with eight components. 3. APPLICATIONS We will show that our result for the first-order system can be used to obtain Carleman estimates for the stationary and dynamic system of elasticity. These Carleman estimates will not only bound the displacement vector but also its firstorder derivatives. REFERENCES [1] Calderón, A. (1959): Uniqueness in the Cauchy problem for partial differential equations. Amer. J. Math. vol. 80, 16-36 [2] Eller, M., Isakov, V., Nakamura, G. and Tataru, D. (2002): Uniqueness and stability in the Cauchy problem for Maxwell and the elasticity system. Studies in Mathematics and its Applications vol. 31 D. Cioranescu and J.L. Lions (editors), 329-349 [3] Imanuvilev, O. and Yamamoto, M. 2004: Carleman estimates for a stationary isotropic Lamé system and its applications. Applicable Analysis vol.83, 243-270