Siberian Mathematical Journal

, Volume 47, Issue 1, pp 135–151

Carleman Estimates for Second-Order Hyperbolic Equations

  • V. G. Romanov

DOI: 10.1007/s11202-006-0014-9

Cite this article as:
Romanov, V.G. Sib Math J (2006) 47: 135. doi:10.1007/s11202-006-0014-9


In the space of variables (x, t) ∈ ℝn+1, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ⊂ ℝn+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) = s2(x, x0) − pt2, where s(x, x0) is the distance between points x and x0 in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x0 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 k0 ≥ 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 k0 and supx∈Ωs2(x, x0) satisfies some smallness condition.


Carleman estimate Cauchy problem stability uniqueness 

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. G. Romanov
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia