By Feruccio Colombini, Claude Zuily
The articles during this quantity mirror a next improvement after a systematic assembly entitled Carleman Estimates and keep an eye on thought, held in Cartona in September 1999. The 14 research-level articles, written through specialists, specialize in new effects on Carleman estimates and their purposes to strong point and controlla bility of partial differential equations and platforms. the most issues are specified continuation for elliptic PDEs and platforms, con trol thought and inverse difficulties. New effects on powerful forte for moment or greater order operators are explored intimately in different papers. within the quarter of keep an eye on conception. the reader will locate functions of Carleman estimates to stabiliza tion, observability and designated keep watch over for the wave and the SchrOdinger equations. a last paper offers a difficult record of open difficulties concerning keep watch over lability of linear and sernilinear warmth equations. The papers comprise exhaustive and primarily self-contained proofs at once ac cessible to mathematicians, physicists, and graduate scholars with an undemanding historical past in PDEs. individuals 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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Extra info for Carleman Estimates and Applications to Uniqueness and Control Theory (Progress in Nonlinear Differential Equations and Their Applications)
Our approach, consisting of diagonalizing the system near the boundary is the main technical part of this work. 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.
Melrose. Singularities and energy decay in acoustical scattering. Duke Math. J. 46 (1979). 43—59.  P. Morse and H. Feshbach. Methods of Theoretical Physics. McGrawHill. New York. 1953. B. Meirose and 3. Sjöstrand. Singularities of boundary value problems I, Comm Pure Appi. Math. 31 (1978), 593-617. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, comm. Pure Appl. Math. 28 (1975), 229—264. V. Ralston. Solutions of the wave equation with localized energy.
N); moreover the eigenvalues of the principal symbol of N have negative imaginary parts. 2 — A)]. 2. Let 1Z = diag(O. e(x,(r)) and we have C< i) ii) + > Cdiag(0. Id) in on = 0} n suppXo. 12) M. 10) we have = —21m(op(1Z)op(fl)w, w) + 21m(op(R)w. F) + )w, w). 1). 2 r) First we prove that for any (x. the restriction 8+ of B in r) is an isomorphism. The eigenvalues of A are = with multiplicity respectivly (n — = —irp' ± and Now let X = (X1, X2) E 20; then X satisfies 1) and 1. Xi=0. ,n_2} 418 ) . 21) 0.
Carleman Estimates and Applications to Uniqueness and Control Theory (Progress in Nonlinear Differential Equations and Their Applications) by Feruccio Colombini, Claude Zuily