---
tag: 269C7M56G5
kind: statement
role: theorem
name: Ball's existence theorem in nonlinear elasticity
msc: [74B20, 49J10, 74G65]
uses: [8VYYWQ5CMB, M4CEZ5K9HP, 00KDZRW4C0, 0XPKR18APA, TE379HBTY0]
about: [8VYYWQ5CMB]
sources:
  - url: https://en.wikipedia.org/wiki/Polyconvex_function
---

## Statement

Let $\Omega \subseteq \R^3$ be a bounded open connected set with Lipschitz boundary, and let $\Gamma_0 \subseteq \partial \Omega$ have positive surface measure.
Let $\mathbb{M}^3_+$ be the set of $3 \times 3$ real matrices with positive [[determinant|00KDZRW4C0]], and let $W \colon \Omega \times \mathbb{M}^3_+ \to \R$ be a stored energy function such that:

1. ([[polyconvexity|8VYYWQ5CMB]]) $W(x, F) = G(x, F, \operatorname{cof} F, \det F)$ for a function $G \colon \Omega \times \R^9 \times \R^9 \times (0, \infty) \to \R$ that is convex in its last three arguments for almost every $x$, and is a Carathéodory function;
2. (behaviour under compression) $W(x, F) \to +\infty$ as $\det F \to 0^+$, for almost every $x$;
3. (coercivity) $W(x, F) \ge \alpha \big( \abs{F}^p + \abs{\operatorname{cof} F}^q + (\det F)^r \big) + \beta$ for constants $\alpha > 0$, $\beta \in \R$, $p \ge 2$, $q \ge p / (p - 1)$ and $r > 1$.

Here $\operatorname{cof} F$ is the [[cofactor|0XPKR18APA]] matrix.
Let $\varphi_0 \colon \Gamma_0 \to \R^3$ be given.
Let $\mathcal{A}$ be the set of $\varphi$ in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\Omega; \R^3)$ with $\varphi = \varphi_0$ on $\Gamma_0$ such that:
$\operatorname{cof} \nabla \varphi \in L^q$, $\det \nabla \varphi \in L^r$, and $\det \nabla \varphi > 0$ almost everywhere.
Let $I(\varphi) = \int_\Omega W(x, \nabla \varphi(x)) \, dx$.
If $\inf_{\mathcal{A}} I < +\infty$, then there is $\varphi \in \mathcal{A}$ with $I(\varphi) = \inf_{\mathcal{A}} I$.

## Remarks

The proof uses the [[direct method|TE379HBTY0]].
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 $p = q = 2$.

## Explanations

### L1 human

An elastic body, clamped on part of its boundary, settles into a shape of least stored energy.
Ball's theorem says that such a shape exists for a large class of realistic materials.
The materials may resist changes of length, of area and of volume, each in a convex way, and they resist being crushed to zero volume infinitely strongly.

### L2 human

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.