A notion of polyconvex function on a surface suggested by nonlinear shell theory

Combining the definitions set forth by J. Ball in 1977 and by J. Ball, J.C. Currie, and P.J. Olver in 1981, we propose in this Note a definition of a “polyconvex function on a surface”. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, which eventually allows to establish the existence of minimizers.