The complementing condition (CC) is an algebraic compatibility requirement between the principal part of a linear elliptic partial differential operator and the principal part of the corresponding boundary operators. When the CC holds the linear boundary value problem has many important functional analytic properties. Recently it has been found that the CC plays a very important role in the construction of a generalized degree, with all the properties of the Leray-Schauder degree, applicable to a general class of problems in nonlinear elasticity. In this paper we discuss the role of the CC in such development and present some examples of boundary value problems in nonlinear elasticity to which this new degree is applicable. In addition, when the CC fails we present some recent results on the implications of this failure in the context of nonlinear elasticity.

The work of Negrón-Marrero was sponsored in part by the University of Puerto Rico at Humacao (UPRH) during a sabbatical leave, and by the NSF-PREM Program of the UPRH (Grant No. DMR-0934195).
The work of Montes-Pizarro was sponsored in part by the Institute for Interdisciplinary Research at the University of Puerto Rico at Cayey as matching funds to the RIMI-NIH program (Grant No. GM-63039-01).