Resolvent estimates for the Lamé operator and failure of Carleman estimates

Abstract

In this paper, we consider the Lamé operator −∆ * and study resolvent estimate, uniform Sobolev estimate, and Carlman estimate for −∆ * . First, we obtain sharp L p -L q resolvent estimates for −∆ * for admissible p, q. This extends the particular case q = p p−1 due to Cossetti [6]. Secondly, we show failure of uniform Sobolev estimate and Carlman estimate for −∆ * . Our approach is different from that in [6] which relies heavily on the Helmholtz decomposition. Instead, we directly analyze the Fourier multiplier of the resolvent. This direct approach 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 −∆ * . Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator −∆ * even though the uniform resolvent estimates for −∆ * are valid for certain range of p, q. This contrasts with the classical result regarding the Laplacian ∆ due to Kenig, Ruiz, and Sogge [21] in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carlman estimates for ∆. We also describe locations of the L q -eigenvalues of −∆ * + V with complex potential V by making use of the sharp L p -L q resolvent estimates for −∆ * .

Strichartz and uniform Sobolev inequalities for the elastic wave equation

“…So, the two are more or less equivalent. However, three of the authors [22] recently proved that if is replaced by the (d-dimensional) Lamé operator ∆ * R d in the above ((1.8) and (1.9)), then the uniform Sobolev inequality (1.8) and Carleman inequality (1.9) fail, while the uniform (and even non-uniform sharp) resolvent estimates are available in the general context of [21] (also, see [2,7]). This shows a fundamental difference between ∆ * and ∆.…”

“…See Corollary 6.1 in Section 6. In view of [22], if the first order spatial derivatives ∇ x is involved in the right side of (1.10), it seems that any uniform estimate cannot hold true. However, we do not pursue this issue here.…”

Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case

“…Instead we choose to diagonalize the Fourier multiplier to get into the position to use resolvent estimates for the fractional Laplacian. Kwon-Lee-Seo [16] previously used a diagonalization to prove resolvent estimates for the Lamé operator. In three spatial dimensions we consider µ > 0 and ε = diag(ε 1 , ε 2 , ε 3 ).…”

“…The short argument is standard by now (cf. [15,16]), but contained for the sake of completeness. Let u ∈ L q 0 (R d ) be an eigenfunction of P + V with eigenvalue E ∈ C\R and suppose that E ∈ Z p,q (ℓ).…”

“…We refer to the works [3,12] for context. Finally, it would be interesting to investigate L p -L q -Carleman estimates, which were shown to fail for the Lamé operator in [16].…”