Resolvent estimates for the Lamé operator and failure of Carleman estimates

2019/12/29 by Kwon, Yehyun, Lee, Sanghyuk, Seo, Ihyeok
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.12620

Abstract

In this paper, we consider the Lamé operator -Δ^∗ and study resolvent estimate, uniform Sobolev estimate, and Carleman estimate for -Δ^∗. First, we obtain sharp L^p--L^q resolvent estimates for -Δ^∗ for admissible p,q. This extends the particular case q=\frac pp-1 due to Barceló et al. \citeBFPRV and Cossetti \citeCo19. Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for -Δ^∗. For the purpose we directly analyze the Fourier multiplier of the resolvent. This allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp L^p--L^q bounds for the resolvent of -Δ^∗. Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator -Δ^∗ even though the uniform resolvent estimates for -Δ^∗ are valid for certain range of p, q. This contrasts with the classical result regarding the Laplacian Δ due to Kenig, Ruiz, and Sogge \citeKRS87 in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carleman estimates for Δ. We also describe locations of the L^q-eigenvalues of -Δ^∗+V with complex potential V by making use of the sharp L^p--L^q resolvent estimates for -Δ^∗.

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