B. Valette - Les oscillations gravito-élastiques de la Terre une invitation à l'étude d'une classe de problèmes spectraux

Internal geophisycs.

Important:

• average density of earth: 5.5, then, we move drastically to 10. Geophysical motivations

• The exterior is considered as solid, and the interior is considered as a liquid, with a big all in the liquid part. Consider that the equilibrium of the ball solid is hydrostatic.

• suppose that we are in a configuration of stationary equilibrium, and suppose an adiabatic evolution. Then, linearisation of first order of the equation. Main idea: perturbation of strains, perturbation of density, of the field of gravity

• Obtains also the equations of motion for the core, liquid.

• Then, defines the interface solid-liquid equations.

• Finally, computes the dynamic evolution in the rotational configuration, with an external excitation: spectral excitation. The excitation leads to generalised spectral problems

• $\mathcal{X}$: Banach space

• para-closed space: $D\subseteq \mathcal{X}$: $D$ is injected continuously within $\mathcal{X}$ (=$D\mapsto \mathcal{X}$ is continuous). The paraclosed topologie is unique.

• Consider $l:D\to \mathcal{X}$ closed: it means that its graph is closed in $\mathcal{X}\times \mathcal{X}$.

• A closed space + a borned space is always a closed space

• Define the Fredholm operator and the semi-Fredholm operator

• Two key results Consider a closed operator

  • Generalised Weil criterion
  • Schur decomposition If, in a Banach space, if we have the luck of having a decomposition of $D=D_1\oplus D_2$ and $\mathcal{X}={\mathcal{X}}_1\oplus {\mathcal{X}}_2$, this leads that the operator $L$ can be decomposed as a multiplication of two triangle matrices and a diagonal matrices.

• quotient topology

• Definition of the operator of elasto-gravity, using the representation theorem of Kato