Well-posedness, regularity, asymptotic behavior and analyticity for some plate-membrane type transmission problems

In this thesis some plate-membrane type transmission problems are studied. Three dampings are considered on the structure: thermal and structural for the plate, and global viscoelastic of Kelvin-Voigt type on the membrane. Sometimes some damping is removed from the structure. The plate may or may not have an inertial term. In the presence and/or absence of any of the elements mentioned above, we establish existence and uniqueness of solution of the system, which depends continuously on the initial data. We also obtain results of regularity, stability and analyticity. We use the semigroup approach to show the well-posedness our system. Following an idea of proof of regularity developed by Avalos and Lasiecka, we prove that if the inertial term is present or absent then the boundary and transmission conditions hold in the strong sense of the trace when the initial data are smooth enough. Then, using a general criteria of Arendt-Batty, we show the strong stability of our system when the membrane is damped and the plate is with or without rotational inertia. Employing a spectral approach, we indirectly prove exponential stability when the plate has rotational inertia and the structure is totally damped. This asymptotic behavior of the solutions is lost when we remove the viscoelastic component of the membrane. Under this situation, we impose a geometrical condition on the membrane boundary and obtain that the solutions decay polynomially with a rate of order at least 1/25 when the plate has rotational inertia and structural damping. Finally, using a well-known Liu-Zheng criterion we prove by contradiction the analyticity of the system when the membrane has Kelvin-Voigt damping and the thermoelastic plate is considered without inertial term and without structural damping.DoctoradoDoctor en Ciencias Naturale

Espacios de Besov sobre el toro n-dimensional Tn e inmersiones continuas

Este trabajo tiene como objetivo el estudio de las propiedades de los espacios de Besov toroidales o periódicos Banach valuados, las inmersiones entre ellos y las inmersiones de estos espacios en espacios de Sobolev periódicos Banach valuados. Estos resultados, que en su mayoría se encuentran en el Capítulo 4, generalizan resultados análogos al caso del toro unidimensional tratados en \cite{Arendt and Bu}. Al parecer, en la literatura existente (al menos en lo que nuestra búsqueda indica) no se encuentran resultados como los que presentamos en el último capítulo de este trabajo sobre los espacios de Besov B^s_{p, q}(T^n, E), cuando n es mayor o igual a 1 y E es un espacio de Banach arbitrario.MaestríaMagister en Matemática

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and <i>C</i>^∞ Semigroups

We consider Fourier multiplier systems on Rn with components belonging to the standard Hörmander class S1,0mRn, but with limited regularity. Using a notion of parameter-ellipticity with respect to a subsector Λ⊂C (introduced by Denk, Saal, and Seiler) we show the generation of both C∞ semigroups and analytic semigroups (in a particular case) on the Sobolev spaces WpkRn,Cq with k∈N0, 1≤p∞ and q∈N. For the proofs, we modify and improve a crucial estimate from Denk, Saal and Seiler, on the inverse matrix of the symbol (see Lemma 2). As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces

Long-time asymptotics for a coupled thermoelastic plate-membrane system

publishe