Normal hyperbolicity of center manifolds and Saint-Venant's principle (Q918767)

The theory of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations for elliptic problems in cylindrical domains. It is shown that the theory of normal hyperbolicity and the theory of Saint-Venant's principle in elastostatics are two different aspects of the same mathematical concept. A reformulation of nonlinear elliptic problems in cylindrical domains to a formal evolution, where the axial variable playing the role of time is given to use the theory of infinite-dimensional center manifolds. It is proven that all solutions u(t) staying close to the center of manifold satisfy a Saint-Venant type estimation that is \(\| u(t)- \tilde u(t)\| \leq C\exp (-\alpha (\ell -| t|))\) for \(t\in (- \ell,\ell)\), where C and \(\alpha\) are independent of \(\ell\), and \(\tilde u\) is a solution on the center manifold. The end of the paper deals with the classical Saint-Venant principle for three dimensional long nonlinear elastic prismatic bodies in detailed form.