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Quantitative estimates of strong unique continuation for wave equations

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Abstract

The main results of the present paper consist in some quantitative estimates for solutions to the wave equation \(\partial ^2_{t}u-\text{ div }(A(x)\nabla _x u)=0\). Such estimates imply the following strong unique continuation properties: (a) if u is a solution to the the wave equation and u is flat on a segment \(\{x_0\}\times J\) on the t axis, then u vanishes in a neighborhood of \(\{x_0\}\times J\). (b) Let u be a solution of the above wave equation in \(\Omega \times J\) that vanishes on a a portion \(Z\times J\) where Z is a portion of \(\partial \Omega \) and u is flat on a segment \(\{x_0\}\times J\), \(x_0\in Z\), then u vanishes in a neighborhood of \(\{x_0\}\times J\). The property (a) has been proved by Lebeau (Commun Partial Differ Equ 24:777–783, 1999).

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Acknowledgments

The paper was partially supported by GNAMPA-INdAM.

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  1. Università degli Studi di Firenze, Florence, Italy

    S. Vessella

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Vessella, S. Quantitative estimates of strong unique continuation for wave equations. Math. Ann. 367, 135–164 (2017). https://doi.org/10.1007/s00208-016-1383-4

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  • DOI: https://doi.org/10.1007/s00208-016-1383-4

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