Abstract
In a domain D = Ω × (−T,T) we consider a differential inequality whose left-hand side contains a linear second-order hyperbolic operator with coefficients depending only on x ∈ ℝ n, n ≥ 2, and the right-hand side contains the modulus of the gradient of the sought function. We supplement the inequality with the Cauchy data on the lateral part of the boundary of D and consider the problem of estimating a solution to the differential inequality satisfying the Cauchy data. We establish the estimate under some assumptions that involves the upper bound of the sectional curvatures of the Riemannian space associated with the differential operator, the Riemannian diameter of Ω, and the length of the interval (−T,T). The result is generalized to the case of compact domains bounded from above and below by characteristic surfaces.
Similar content being viewed by others
References
Shishatskii S. P., “A priori estimates in the problem of extending a wave field from a time-like cylindrical surface,” Dokl. Akad. Nauk SSSR, 213, No. 1, 49–50 (1973).
Lavrent’ev M. M., Romanov V. G., and Shishatskii S. P., Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980).
Klibanov M. V. and Malinsky J., “The Newton-Kantorovich method for the three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data,” Inverse Problems, 7, No. 4, 577–596 (1991).
Klibanov M. V. and Timonov A., Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, The Netherlands (2004).
Imanuvilov O. Yu. and Yamamoto M., “Global Lipschitz stability in an inverse hyperbolic problem by interior observations,” Inverse Problems, 17, No. 4, 717–728 (2001).
Carleman T., “Sur un problème d’unicité pourles systèmes d’équetions aux dérivées partielles à deux variables indépendentes,” Ark. Mat. Ser. B. Astr. Fys., 26, No. 17, 1–9 (1939).
Isakov V., “Carleman estimates and applications to inverse problems,” Milan J. Math., 72, 249–271 (2004).
Lasiecka I., Triggiani R., and Yao P. F., “Inverse/observability estimates for second order hyperbolic equations with variable coefficient principal part and first-order terms,” J. Math. Anal. Appl., 235, 13–57 (1999).
Triggiani R. and Yao P. F., “Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot,” Appl. Math. Optim., 46, No. 2/3, 334–375 (2002).
Romanov V. G., “Carleman estimates for second-order hyperbolic equations,” Siberian Math. J., 47, No. 1, 135–151 (2006).
Romanov V. G., Stability in Inverse Problems [in Russian], Nauchnyĭ Mir, Moscow (2005).
Romanov V. G., “A stability estimate for a solution to the wave equation with the Cauchy data on a timelike cylindrical surface,” Siberian Math. J., 46, No. 5, 925–934 (2005).
Author information
Authors and Affiliations
Additional information
Original Russian Text Copyright © 2006 Romanov V. G.
The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00171) and the Program “ Universities of Russia” (Grant UR.04.01.200).
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 3, pp. 626–635, May–June, 2006.
Rights and permissions
About this article
Cite this article
Romanov, V.G. Estimates for a solution to one differential inequality. Sib Math J 47, 517–525 (2006). https://doi.org/10.1007/s11202-006-0063-0
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-006-0063-0