Ball's existence theorem in nonlinear elasticity

Remarks

The proof uses the direct method. Coercivity bounds a minimising sequence $\varphi_k$, so along a subsequence $\varphi_k$, $\operatorname{cof} \nabla \varphi_k$ and $\det \nabla \varphi_k$ converge weakly in $L^p$, $L^q$ and $L^r$. The key point is that the weak limits are $\operatorname{cof} \nabla \varphi$ and $\det \nabla \varphi$ of the limit map, because minors are null Lagrangians in divergence form. Convexity of $G$ then gives lower semicontinuity, and condition 2 keeps $\det \nabla \varphi > 0$ almost everywhere.

The theorem is due to John Ball, in 1977, and it holds with dead loads added to $I$ as linear terms. Convexity of $W$ in $F$ would be an unsuitable hypothesis, since the set $\mathbb{M}^3_+$ is not convex and $W$ must blow up as $\det F \to 0^+$.

Example: the energies $W(F) = a \abs{F}^2 + b \abs{\operatorname{cof} F}^2 + \Gamma(\det F)$, with $a, b > 0$ and $\Gamma$ convex on $(0, \infty)$ with $\Gamma(d) \to +\infty$ as $d \to 0^+$ and $\Gamma(d) \ge c \, d^r - C$ for some $c > 0$, $C$ and $r > 1$, satisfy the hypotheses with $p = q = 2$.

Going deeper

For a student

Before this theorem, existence in nonlinear elasticity was known mainly near the reference configuration, by the implicit function theorem. Ball's idea was to replace convexity, which is physically wrong, by polyconvexity, and to use the weak continuity of determinants and cofactors. The minimiser is only a Sobolev map, and whether it satisfies the equilibrium equations, is injective, or is smooth are difficult further questions. The theorem made polyconvexity a central notion of the calculus of variations, next to quasiconvexity and rank-one convexity.