By Feruccio Colombini, Claude Zuily

The articles during this quantity mirror a next improvement after a systematic assembly entitled Carleman Estimates and keep watch over concept, held in Cartona in September 1999. The 14 research-level articles, written by means of specialists, specialise in new effects on Carleman estimates and their purposes to forte and controlla bility of partial differential equations and platforms. the most subject matters are certain continuation for elliptic PDEs and structures, con trol concept and inverse difficulties. New effects on powerful specialty for moment or greater order operators are explored intimately in different papers. within the quarter of keep watch over thought. the reader will locate purposes of Carleman estimates to stabiliza tion, observability and specific keep an eye on for the wave and the SchrOdinger equations. a last paper provides a hard record of open difficulties related to keep an eye on lability of linear and sernilinear warmth equations. The papers include exhaustive and primarily self-contained proofs at once ac cessible to mathematicians, physicists, and graduate scholars with an straight forward historical past in PDEs. members are L. Aloui, M. Bellassoued, N. Burq, F. Colombini, B. Dehman, C. Grammatico, M. Khenissi, H. Koch, P. Le Borgne, N. Lerner, T. Nishitani. T. Okaji, K.D. Phung, R. Regbaoui, X. Saint Raymond, D. Tataru, and E. Zuazua.

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Vodev. Neumann resonances in linear elasticity for an arbitrary body, Comm. in Math. Phy. 176 (1996). 645—659. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures AppI. 9(4), 75 (1996), 367—408. [21] M. Taylor. Rayleigh waves in linear elasticity as a propagation of singularities phenomenon, in Proceedings of the Conference on Partial Duff. Equa. Ceo.. Marcel Dekker, New York, 1979, pp. 273—291. [22] M. Taylor, Reflection of singularities of solution to systems of differential equations, Comm.

In this paper we prove strong uniqueness results which are not included in these theorems. 50 F. Colombini, C. Cranunatico Finally let us recall that Wolff [11] has shown that, for n 4, there is a not identically zero, such that, for a certain constant C,

1 Reduction of the problems Reduction of the Laplacian be a bounded smooth domain of R" with boundary 0f10 of class Ccc. 0, we denote by x = the system of normal geodesic coordinates where x' E and E IR are characterized by Let = = > 0}; dist(x',x) = dist(x,Oflo). Carleman Estimate and Decay Rate 25 In this system of coordinates the principal symbol of the Laplace operator takes the form = tt(x + r(x. 3) is a quadratic form, such that there exist for C > where r(x. 0, T(8ft0). for any x E K. 5) then we have = = 0.