Proposition

Let $I$ be an interval. Let $M$ be a collection of measurable functions $\mu$ such that

Then there exists a constant $\varepsilon$, which depends only on the length of $I$, but not on $\mu$, such that

Proof.

In this proof $C$ denotes a positive constant which depends on the length of $I$, but not on $\mu$.

The hypothesis implies that $\|\mu\|_{H^1(I)}\le C$. Moreover, since $\int_I \mu^{-2}{\rm d}x<+\infty$, we have

We claim that for each $\mu$ there is a constant $\varepsilon$ such that

By contradiction, suppose that $\eqref{eq:2}$ does not hold true. Then $\mu(x_0)=0$ for some $x_0$. By Morrey’s inequality, we have $\|\mu\|_{C^{0,1/2}(\overline I)}\le C$. Accordingly, for $x-x_0$ we have

for all $x\in I$. Thus

This yields a contradiction.

We now claim that the constant $\varepsilon$ in can be take to be the same for all $\mu$.

Assume by contradiction that $\inf_{\mu\in M}\varepsilon(\mu)=0$. Then there exists a sequence $x_n$ in $\overline I$ and a sequence $\mu_n$ in $M$ such that

Then $x_n\to x$ (since $\overline I$ is compact). The sequence $\mu_n$ converges weakly to some limit $\mu$ in $H^1(I)$. By the Rellich-Kondrachov theorem, $C(\overline I)$ is compactly embedded in $H^1(I)$, thus the sequence $\mu_n$ converges uniformly. By Fatou’s lemma,

Thus, $\mu\in M$ and hence $\varepsilon(\mu)>0$. Since the convergence of $\mu_n$ to $\mu$ is uniform, we have $\mu_n(x_n)\to\mu(x)\ge \varepsilon(\mu)>0$, which contradicts $\eqref{eq:6}$.

Remark Note carefully that continuity of $\mu$ alone without the integrability of $1/\mu^2$ would not suffice to guarantee $\eqref{eq:2}$.

References

This proof is an adaptation from Theorem 2.5.3 of the book:

• Kružík, M., & Roubíček, T. (2019). Mathematical Methods in Continuum Mechanics of Solids. Springer International Publishing. https://doi.org/10.1007/978-3-030-02065-1,

which in turn derives from a result first proved in the following paper

• Healey, T. J., & Krömer, S. (2009). Injective weak solutions in second-gradient nonlinear elasticity. ESAIM: Control, Optimisation and Calculus of Variations, 15(4), 863–871. https://doi.org/10.1051/cocv:2008050

This result provide conditions for having a positive uniform lower bound on the determinant of a family of minimizers from linear elastostatics for a non-simple hyperelastic material.