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Generalizations of the Lamé-Navier operator in Clifford analysis
Clifford Analysis has helped to effectively interpret many of the equations of Mathematical
Physics, and in particular of the Mechanics of Continuous Media. In this paper we study a
natural generalization of the classical Lamé-Navier operator on Clifford algebras. The use of
Dirac operators constructed with arbitrary orthonormal bases leads to a great variety of
systems of partial differential equations of mathematical and physical interest. First, several
essential properties such as invariance over k-vector fields and ellipticity are studied. In
addition, a rewriting of the Lamé-Navier system in terms of the longitudinal and transverse
modules is presented. Finally, the Dirichlet problem associated with functions that cancel the
generalized Lamé-Navier operator is considered, and we determine the condition that causes
the ill-posedness of problem in the Hadamard sense.
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