Unique Continuation and an Inverse Problem for Hyperbolic Equations across a General Hypersurface

For a hyperbolic equation p(x, t)∂ t u(x, t) = ∆u(x, t)+ ∑n j=1 qj(x, t)∂ju+ qn+1(x, t)∂tu+r(x, t)u in R n×R with p ∈ C and q1, ...., qn+1, r ∈ L∞, we consider the unique continuation and an inverse problem across a non-convex hypersurface Γ. Let Γ be a part of the boundary of a domain and let ν(x) be the inward unit normal vector to Γ at x. Then we prove the unique continuation near a point x0 across Γ if ∇p(x0, t) · ν(x0) < 0. Moreover we establish the conditional stability in the continuation. Next we prove the conditional stability in the inverse problem of determining a coefficient r(x) from Cauchy data on Γ over a time interval. The key is a Carleman estimate in level sets of paraboloid shapes. §