A spectral study of an infinite axisymmetric elastic layer (Q2781515)

The paper extends previous results obtained by the author concerning the formal properties associated to eigenvalues (existence, etc.), i.e. of displacements travelling without source and with energy localized close to the symmetry axis of the layer, for Lamé coefficients depending also upon the transversal coordinates. For this purpose is defined the corresponding two dimensional problem with differential operators and the variational one with other operators which as further stated leads to self-adjoint operators occuring in similar differential equations. The analysis matters for the construction of Green functions which enable to solve transient problems of waves with source. The demonstration is performed by establishing the coerciveness of the introduced bilinear form intervening in formulations using Korn's inequality and Kato's first representation theorem. Based on these results it follows that the occuring spectrum is defined on the positive part of \(\mathbb{R}\), being composed by a continuous (essential) and discrete spectrum. The lower bounds of the first one is given by the spectrum of an homogeneous layer as demonstrated by means of a compactness lemma for weighted Sobolev spaces. The discrete (point) spectrum is not empty if the transversal modulus satisfies a determined inequality and that is the final theoretical result.