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Lq-solutions to the Cosserat spectrum in bounded and exterior domains
(2005)
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Stephan Weyers
- In the present paper we consider the existence of non-trivial classical and weak Lq-solutions of the Cosserat spectrum $$ Delta U u = a abla Div U u, qquad U u Big|_{partial G}=0 $$ where G is a bounded or an exterior domain with sufficiently smooth boundary. This problem firstly was investigated by Eugene and Francois Cosserat. It is a special case of the Lame equation and describes the displacement of a homogeneous isotropic linear static elastic body without exterior forces. We can prove that a = 1 is an eigenvalue of infinite multiplicity and a = 2 is an accumulation point of eigenvalues of finite multiplicity. E. and F. Cosserat (1900) studied the classical Cosserat spectrum for certain types of domains like a ball, a spherical shell or an ellipsoid. General results are due to Mikhlin (1973), who investigated the Cosserat spectrum for n=3 and q=2, and Kozhevnikov (1993), who treated bounded domains in the case n=3 and q=2. Kozhevnikovs proof is based on the theory of pseudodifferential operators. Faierman, Fries, Mennicken and Möller (2000) gave a direct proof for bounded domains, n>=2 and q=2. Michel Crouzeix 1997 gave a simple proof for bounded domains, n=2,3 and q=2. In this paper we use the idea of Crouzeix to prove the results for bounded and exterior domains, n>=2 and 1<q. For the Lq-solutions of eigenvalues a in R {1,2} we can prove the existence of higher (classical) derivatives. Furthermore they do not depend on q. a =2 is an accumulation point of eigenvalues of the classical Cosserat spectrum, too, and a =1 is also a classical eigenvalue. As an approach we searched for a relationship of Greens function of the Laplacian to the reproducing kernel in Bergman spaces. We couldnt prove that directly. But after solving the Cosserat spectrum in another way we can prove the relationship indirectly.
