Carleman Estimates for Second-Order Hyperbolic Equations

Abstract

In the space of variables (x, t) ∈ ℝ n+1, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ⊂ ℝ 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(2τϕ(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 ϕ(x, t) to be the function ϕ(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.