On Constanda's matrix of nonuniqueness in a theory of plates in asymmetric elasticity
We apply the direct method to the solution of the Dirichlet problem in the bending of micropolar plates and show that a necessary and sufficient condition for solubility is the nonsingularity of a special constant matrix constructed for any smooth closed boundary curve. Introduction. In the application of the direct method to the solution of the Dirichlet problem for the two-dimensional Laplace equation, there are certain boundary curves (of logarithmic capacity one [1]) for which the corresponding homogeneous Fredholm integral equation of the first kind has nonzero solutions. This leads ultimately to the breakdown of the method [2]. In this paper, proceeding as in [3], we show how a similar situation occurs when we apply the direct method to the solution of the Dirichlet problem for the equations of bending of micropolar plates [4], We identify the particular boundary curves using a generalisation of the concept of capacity in which Robin's constant is replaced by a constant (3 x 3)-matrix whose nonsingularity is used to express a necessary and sufficient condition for the solubility of the Dirichlet problem by the direct method. Preliminary results. Throughout the paper we assume that, unless otherwise stated, Greek and Latin indices take the values 1, 2 and 1, 2,3 respectively, we sum over repeated indices, M.pxq is the space of (p x (^-matrix functions, are the columns of a matrix H £ Mpxq, and En is the unit matrix in AinxnFinally, if X is a space of scalar functions and Q G Mpxq, then Q £ X means that every component of Q belongs to X. We consider a homogeneous and isotropic micropolar plate occupying the region S x [—ho/2, ho/2], where S C R2 is a domain bounded by a simple closed C2-curve dS and h0 = const << diamS" is the thickness. In the absence of body forces and couples and of forces and couples on the faces, the equilibrium equations for bending can be written Received April 10, 1995. 1991 Mathematics Subject Classification. Primary 35J55, 45E05, 73C35. ©1998 Brown University 137 138 CHONG-QING RU and PETER SCHIAVONE