Theory of Affine Shells: Higher Order Estimates

Abstract

The classical Theory of Shells has been exposed, through the contributions of many authors, within the framework of Euclidean Geometry, i.e., based on the classical theory of surfaces in three-dimensional space, which is invariant under translations and rotations. For diverse viewpoints of presentation see (F. John, Comm. Pure Appl. Math. 18(1/2): 235-267 (1965); 24(5): 583-615 (1971)) and other references therein. More recently, we ourselves have been working in a new development of this theory based, from the geometrical viewpoint, in those objects which remain invariant under the action of the Unimodular Affine Group, i.e., dealing with Affine Surface Geometry (S. Gigena et al., Mec. Comp., 21: 18621881 (2002); 22: 1953-1963 (2003); 23: 639-652 (2004); 24: 2745-2758 (2005)). In this paper we study exclusively the behavior of physical objects of the shell in the interior, without reference to any boundary conditions at the edge. For the interior behavior one needs as the only tool a certain kind of a priori estimates. These interior regularity estimates, similar to those occurring in the theory of Partial Differential Equations, rigorously assign a definite order of magnitude to every quantity occurring in the theory. Our main goal here is to establish those estimates for the strain and stress tensors, as well as for the higher order covariant derivatives of both, within the framework of the Theory of Affine Shells.