Abstract
The simplest theories for quasilinear elliptic equations for an unknown scalar-valued function u are based upon coercivity conditions that require some energy-like term to exceed const |u|P where p > 1. There are many physical applications of such theories. Theories for quasilinear elliptic systems for a vector-valued u can likewise be based on analogous coercivity conditions. If, however, the system is to consist of the equilibrium equations of nonlinear elasticity with u(z) denoting the position of a material point z, then this kind of coercivity condition is singularly inadequate on both physical and mathematical grounds. This paper first describes the mathematical structure of the quasilinear equations of elastostatics, focusing; on the far more intricate kinds of coercivity conditions that are needed to meet reasonable physical restrictions and to promote mathematical analysis. The role of these new coercivity conditions is then illustrated by an analysis of the existence, regularity, and qualitative behavior of solutions of specific boundary value problems.
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