Unique continuation and an inverse problem for hyperbolic equations across a general hypersurface

We consider a hyperbolic equation p(x, t) ?t2u(x, t) = ?u(x, t) + ?k=1n qk(x, t) ?ku + qn + 1(x, t) ?tu + r(x, t)u in n ? with p C1 and q1, ... , qn+1, r L?. Let ? be a part of the boundary of a domain and let ?(x) be the inward unit normal vector to ? at x. Then we prove the conditional stability in the unique continuation near a point x0 across ? if ?p(x0, t) ? ?(x0) < 0 and the radius of the osculating ball at x0 is large for ??p(x0, t) ? ?(x0). Next we prove the conditional stability in the inverse problem of determining a coefficient r(x) from Cauchy data on ? over a time interval. The key is a Carleman estimate in level sets of paraboloid shapes.

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