Quantitative Strong Unique Continuation for Elliptic Operators—Application to an Inverse Spectral Problem

Abstract

Based on the three-ball inequality and the doubling inequality established in [Math. Ann., 2012, 353(4): 1157–1181], we quantify the strong unique continuation established by Koch and Tataru [Comm. Pure Appl. Math., 2001, 54(3): 339–360] for elliptic operators with unbounded lower-order coefficients. We also obtain a quantitative strong unique continuation for eigenfunctions, which we use to prove that two Dirichlet–Laplace–Beltrami operators are gauge equivalent when their corresponding metrics coincide in the neighborhood of the boundary and their boundary spectral data coincide on a subset of positive measure.

Keywords

Elliptic operators / unbounded coefficients / strong unique continuation / three-ball inequality / doubling inequality / eigenfunctions / Laplace–Beltrami operator / gauge equivalent operators / 35B60 / 35J15 / 35R30

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Peking University