A priori estimates of Mizohata-Takeuchi type for the Navier-Lamé operator
Juan Antonio Barceló, Alberto Ruiz, Mari Cruz Vilela, Jim Wright
TL;DR
This work resolves the Mizohata-Takeuchi-type weighted estimates for the resolvent of the Navier-Lamé operator in dimensions $d=2,3$ for radial weights in the class $\\mathcal{T}_{rad}(\\mathbb{R}^d)$, establishing the a priori bounds $\\int_{\\mathbb{R}^d}|\\mathbf{u}|^2 V dx \\le \\frac{c}{\\omega^2}|||V|||^2 \\\int_{\\mathbb{R}^d}|\\mathbf{f}|^2 V^{-1} dx$ and $\\sup_{1\\le j\\le d} \\\int_{\\mathbb{R}^d} | abla u_j|^2 V dx \\le c|||V|||^2 \\\int_{\\mathbb{R}^d}|\\mathbf{f}|^2 V^{-1} dx$, under Kupradze radiation conditions. The radial MT class is shown not to be invariant under the Hardy-Littlewood maximal operator, which prevents a direct extension of Helmholtz-based methods to the Navier-Lamé system; instead, the authors develop new addition formulas for the fundamental solution and delicate cancellation arguments to control the resolvent. The results provide robust tools for boundary and scattering problems in elasticity and pave the way for limiting absorption principle-type conclusions in this setting. The paper also contributes explicit spherical-harmonic decompositions of the Navier-Lamé fundamental solution, which are of independent interest for elasticity theory.
Abstract
The Mizohata-Takeuchi conjecture for the resolvent of the Navier-Lamé equation is a weighted estimate with weights in the so-called Mizohata-Takeuchi class for this operator when one approaches the spectrum (Limiting Absorption Principles). We prove this conjecture in dimensions 2 and 3 for weights with a radial majorant in the Mizohata-Takeuchi class. This result can be seen as an extension of the analogue for the Laplacian given in [8]. We also prove that radial weights in this class are not invariant for the Hardy-Littlewood maximal function, hence the methods in [6] used to extend estimates for the Laplacian to the Navier-Lamé case, do not work.