Unique continuation and an inverse problem for hyperbolic equations across a general hypersurface
AK Amirov, M Yamamoto, J Cheng, M Yamamoto
proceeding (Uluslararası, Hakemli) | Yazar(lar): AK Amirov, M Yamamoto, J Cheng, M Yamamoto | Yıl: 2005 | Dergi/Kurum: SECOND INTERNATIONAL CONFERENCE ON INVERSE PROBLEMS: RECENT THEORETICAL DEVELOPMENTS AND NUMERICAL APPROACHES, 2004 | İndeks: WOS.ISTP | DOI: 10.1088/1742-6596/12/1/001 — Özet: We consider a hyperbolic
We consider a hyperbolic equation p(x,t)partial derivative(2)(t)u(x,t) = Delta u(x,t) + Sigma(n)(k=1) qk(x, t)partial derivative ku+ qn+l (x, t)partial derivative(t)u + r(x, t)u in R-n x R with p is an element of C-1 and q(1)...... q(n+l), r is an element of L-infinity. Let r be a part of the boundary of a domain and let v(x) be the inward unit normal vector to Gamma at x. Then we prove the conditional stability in the unique continuation near a point x(0) across Gamma if del P(x(0), t) (.) v(x(0)) < 0 and the radius of the osculating ball at x(0) is large for -del p(x(0),t) (.) v(x(0)). Next we prove the conditional stability in the inverse problem of determining a coefficient r(x) from Cauchy data on Gamma over a time interval. The key is a Carleman estimate in level sets of paraboloid shapes.