Carleman Estimates for Second-Order Hyperbolic Equations |
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Authors: | V. G. Romanov |
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Affiliation: | (1) Sobolev Institute of Mathematics, Novosibirsk, Russia |
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Abstract: | In the space of variables (x, t) ∈ ? n+1, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? n+1 whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s 2(x, x 0) ? pt 2, where s(x, x 0) is the distance between points x and x 0 in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x 0 can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k 0 ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k 0 and sup x∈Ω s 2(x, x 0) satisfies some smallness condition. |
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Keywords: | Carleman estimate Cauchy problem stability uniqueness |
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