Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains

Abstract

We consider periodic homogenization of nonlinearly elastic composite materials. Under suitable assumptions on the stored energy function (frame indifference; minimality, non-degeneracy and smoothness at identity; $p\ge d$-growth from below), and on the microgeometry of the composite (covering the case of smooth, periodically distributed inclusions with touching boundaries), we prove that in an open neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula that can be represented with help of a corrector. This generalizes a recent result of the authors by significantly relaxing the spatial regularity assumptions on the stored energy function. As an application, we consider the nonlinear elasticity problem for $\varepsilon$-periodic composites, and prove that minimizers (subject to small loading and well-prepared boundary data) satisfy a Lipschitz estimate that is uniform in $0<\varepsilon \ll 1$. A key ingredient of our analysis is a new Lipschitz estimate (under a smallness condition) for monotone systems with spatially piecewise-constant coefficients. The estimate only depends on the geometry of the coefficient’s discontinuity-interfaces, but not on the distance between these interfaces.

A: Appendix

1.1 A.1: Proof of Lemma 6

Proof of Lemma 6

We present the proof only in the case $R=1$, the statement for arbitrary $R>0$ follows by scaling.

Step 1. For all $k\in \mathbb {N}$ there exists $c=c(\beta ,k)<\infty$ such that

$$\begin{aligned} \sum _{\ell =0}^k\Vert {\nabla '}^\ell \nabla u\Vert _{L^2(B_\frac{1}{2})}\le c\Vert \nabla u\Vert _{L^2(B_1)}. \end{aligned}$$

(147)

Indeed, this is routine and follows by differentiating the equation via difference quotients.

Step 2. As in Proposition 1, we set $J_d(u):=({\mathbb {L}}\nabla u)e_d$. Inequality (147) yields corresponding estimates for the derivatives of $J_d(u)$ in $x'$-direction: For every $k\in \mathbb {N}$ there exists $c=c(\beta ,d,k)<\infty$ such that

$$\begin{aligned} \sum _{\ell =0}^{k}\Vert {\nabla '}^\ell J_d(u)\Vert _{L^2(B_{\frac{1}{2}})}\le c \Vert \nabla u\Vert _{L^2(B_1)}. \end{aligned}$$

In order to obtain suitable estimates for $\partial _d J_d(u)$, we use equation (55) in the form of $\partial _d J_d(u)=-\sum _{j=1}^{d-1} \partial _j({\mathbb {L}}\nabla u)e_j$. Hence, we find for every $k\in \mathbb {N}$ a constant $c=c(\beta ,d,k)<\infty$ such that

$$\begin{aligned} \sum _{\ell =0}^k\Vert {\nabla '}^\ell \partial _d J_d(u)\Vert _{L^2(B_{\frac{1}{2}})}\le c \Vert \nabla u\Vert _{L^2(B_1)}. \end{aligned}$$

Step 3. We claim that there exists $c=c(\beta ,d)<\infty$ such that

$$\begin{aligned} \Vert \nabla u\Vert _{L^\infty (B_\frac{1}{2})}\le c\Vert \nabla u\Vert _{L^2(B_1)}. \end{aligned}$$

The following anisotropic Sobolev inequality can be found in [26, Lemma 2.2]: Suppose that ${\nabla '}^\ell f\in L^2(B_1)$ and ${\nabla '}^\ell \partial _d f\in L^2(B_1)$ for all $\ell =0,\dots ,K$ with $\frac{d-1}{2}<K$. Then $f\in C^0(B_1)$ and

$$\begin{aligned} \Vert f\Vert _{L^\infty (B_1)}\le C(d)\sum _{\ell =0}^{K}\left( \Vert {\nabla '}^\ell \partial _d f\Vert _{L^2(B_1)}+\Vert {\nabla '}^\ell f\Vert _{L^2(B_1)}\right) . \end{aligned}$$

(148)

By Step 1 and 2, we can apply (148) to ${\nabla '}u$ and $J_d(u)$ and obtain that there exists $c=c(\beta ,d)<\infty$ such that

$$\begin{aligned} \Vert {\nabla '}u\Vert _{L^\infty (B_{\frac{1}{2}})}+\Vert J_d(u)\Vert _{L^\infty (B_\frac{1}{2})}\le&c\Vert \nabla u\Vert _{L^2(B_1)}. \end{aligned}$$

Finally, the $L^\infty$ estimate for $\partial _d u$ follows from Lemma 4 (with $\mathbf{{a}}(F):={\mathbb {L}}F$). $\square$

1.2 A.2: Proof of Lemma 2

As already mentioned, Lemma 2 is completely standard and well-known, see e.g., [13]. We only provide an argument for the continuity of $D\mathbf{{a}}$ claimed in the second bullet point.

Proof

Throughout the proof we write $\lesssim$ if $\le$ holds up to a multiplicative constant depending only on $\beta$ and d.

Let us first recall that $\mathbf{{a}}_0\in C^1(\mathbb {R}^{d\times d},\mathbb {R}^{d\times d})$ and for every $F,G\in \mathbb {R}^{d\times d}$ it holds

$$\begin{aligned} D\mathbf{{a}}_0(F)[G]:=\int _{Q_1}{\mathbb {L}}_F(G+\nabla \psi _G(x,F))\,dx \end{aligned}$$

where ${\mathbb {L}}_F\in L^\infty (\mathbb {R}^d,\mathbb {R}^{d^4})$ denotes the fourth order tensor satisfying

$$\begin{aligned} {\mathbb {L}}_F(x)G=D\mathbf{{a}}(x,F+\nabla \phi (x,F))[G]\qquad \hbox {for every } G\in \mathbb {R}^{d\times d}\hbox { and a.e.}~x\in \mathbb {R}^d, \end{aligned}$$

and $\psi _G(F)\in H_{{\text {per}}}^1(Q_1)$ is uniquely given by

$$\begin{aligned} {\text {div}}\, {\mathbb {L}}_F(G+\nabla \psi _G(F))=0\quad \hbox {in } {\mathscr {D}}'(\mathbb {R}^d),\qquad (\psi _G(F))_{Q_1}=0. \end{aligned}$$

(149)

The prove of this result can easily deduced from [17, Theorem 5.4] see also [32] (in particular proof of Lemma 3).

Step 1. We claim that there exists $c=c(\beta ,d),q=q(\beta ,d)\in [1,\infty )$ such that for all $F_1,F_2,G\in \mathbb {R}^{d\times d}$ with $|G|=1$ it holds

$$\begin{aligned} \Vert \nabla \psi _G(F_1)-\nabla \psi _G(F_2)\Vert _{L^2(Q_1)}\le c\omega (c|F_1-F_2|)^\frac{1}{q}. \end{aligned}$$

(150)

Substep 1.1. We claim

$$\begin{aligned} \Vert {\mathbb {L}}_{F_2} - {\mathbb {L}}_{F_2}\Vert _{L^1(Q_1)}\lesssim \omega ((1+c)|F_1-F_2|), \end{aligned}$$

(151)

where $c=c(\beta )<\infty$ denotes the constant in (24). Indeed, by (18) and concavity of $\omega$, we obtain

$$\begin{aligned} \Vert {\mathbb {L}}_{F_2} - {\mathbb {L}}_{F_1}\Vert _{L^1(Q_1)}\lesssim&\Vert \omega (|F_1+\nabla \phi (F_1)-(F_2+\nabla \phi (F_2))|)\Vert _{L^1(Q_1)}\\ \le&\omega \left( \Vert F_1+\nabla \phi (F_1)-(F_2+\nabla \phi (F_2))\Vert _{L^1(Q_1)}\right) \end{aligned}$$

and the claim follows by Hölder’s inequality and (24).

Substep 1.2. Conclusion. Equation (149) and Meyers estimate imply that there exists $\mu =\mu (\beta ,d)>0$ such that

$$\begin{aligned} \Vert \psi _G(F)\Vert _{W^{1,2+\mu }(Q_1)}\lesssim |G|=1. \end{aligned}$$

(152)

A combination of (149) and (151) yields

$$\begin{aligned}&\Vert \nabla \psi _G(F_1)-\nabla \psi _G(F_2)\Vert _{L^2(Q_1)}^2\\&\quad \lesssim \int _{Q_1}\langle {\mathbb {L}}_{F_1}(x)(\nabla \psi _G(F_1)- \nabla \psi _G(F_2)),\nabla \psi _G(F_1)- \nabla \psi _G(F_2)\rangle \,dx\\&\quad =\int _{Q_1}\langle ({\mathbb {L}}_{F_2}(x)-{\mathbb {L}}_{F_1}(x))(G+\nabla \psi _G(F_2)),\nabla \psi _G(F_1)- \nabla \psi _G(F_2)\rangle \,dx\\&\quad \le \Vert {\mathbb {L}}_{F_1}-{\mathbb {L}}_{F_2}\Vert _{L^\frac{2(2+\mu )}{\mu }(Q_1)}\Vert G+\nabla \psi _G(F_2)\Vert _{L^{2+\mu }(Q_1)}\Vert \nabla \psi _G(F_1)-\nabla \psi _G(F_2)\Vert _{L^2(Q_1)}. \end{aligned}$$

The claim (150) (with $q=\frac{2(2+\mu )}{\mu }$) follows by Hölders inequality, (151) and (152).

Step 2. Conclusion. Let $F_1,F_2,G\in \mathbb {R}^{d\times d}$ with $|G|=1$ be given. Then, (150) and (151) yield

which proves the claim. $\square$

1.3 A.3: Meyers estimate

In Lemma 8 and Proposition 3, we use a global version of Meyers estimate. For convenience of the reader, we here give a short proof of this well-known result. The key ingredient is the following classic higher integrability result

Theorem 6

([20] Theorem 6.6) Let $K>0$, $m\in (0,1)$, $s>1$ and $B=B_R(x_0)$ for some $x_0\in \mathbb {R}^d$ and $R>0$ be given. Suppose that $f\in L^1(B)$ and $g\in L^s(B)$ are such that for every $z\in \mathbb {R}^d$ and $r>0$ with $B_r(z)\subset B$ it holds

$$\begin{aligned} \Vert f\Vert _{\underline{L}^1(B_\frac{r}{2}(z))}\le K\left( \Vert f\Vert _{\underline{L}^m(B_r(z))}+\Vert g\Vert _{\underline{L}^1(B_r(z))}\right) . \end{aligned}$$

Then there exist $q=q(K,m,s)\in (1,s]$ and $c=c(K,m,s)\in [1,\infty )$ such that $f\in L^q(\frac{1}{2} B)$ and it holds

$$\begin{aligned} \Vert f\Vert _{\underline{L}^q(\frac{1}{2} B)}\le c \left( \Vert f\Vert _{\underline{L}^1(B)}+\Vert g\Vert _{\underline{L}^q(B)}\right) . \end{aligned}$$

Next, we state a well-known global version of Meyer’s estimate.

Lemma 10

Fix $d\ge 2$, $\beta \in (0,1]$ and let $\mathbf{{a}}$ be a coefficient field of class ${\mathcal {A}}_\beta$. Then, for every $p>2$ there exists $p_0=p_0(\beta ,d,p)\in (2,p]$ and $c=c(\beta ,d,p)\in [1,\infty )$ such that if $u\in H_0^1(B)$ and $g\in L^p(B)$, where B denotes either a ball or a cube, satisfy

$$\begin{aligned} {\text {div}}\, \mathbf{{a}}(\nabla u)={\text {div}}\, g\qquad \hbox {in }{\mathscr {D}}'(B), \end{aligned}$$

then

$$\begin{aligned} \Vert \nabla u\Vert _{\underline{L}^{p_0} (B)}\le c \Vert g\Vert _{\underline{L}^{p_0} (B)}. \end{aligned}$$

Proof

Without loss of generality, we consider $B=B_1(0)$, the general case follows by scaling and translation. Throughout the proof, we write $\lesssim$ if $\le$ holds up to a multiplicative constant depending on $\beta$ and d.

We show that for every $z\in \mathbb {R}^d$ and $r>0$ it holds

$$\begin{aligned} \Vert \nabla u\Vert _{\underline{L}^2(B_\frac{r}{2}(z))}\lesssim \Vert \nabla u\Vert _{\underline{L}^{\frac{2d}{d+2}}(B_r(z))}+\Vert g\Vert _{\underline{L}^2(B_r(z))}, \end{aligned}$$

(153)

where we extend u and g by zero on $\mathbb {R}^d\setminus B$. The claim follows by applying Theorem 6.

Following [3, Proposition B.6], we distinguish between the cases $B_r(z)\subset B$ and $B_r(z)\setminus B\ne \emptyset$. Suppose that $B_r(z)\subset B$. Then, a combination of the Caccioppoli inequality and the Sobolev inequality yields

$$\begin{aligned} \Vert \nabla u\Vert _{\underline{L}^2(B_\frac{r}{2}(z))}\lesssim&r^{-1}\Vert u-(u)_{B_r(z)}\Vert _{\underline{L}^2(B_r(z))}+\Vert g\Vert _{\underline{L}^2(B_r(z))}\nonumber \\ \lesssim&\Vert \nabla u\Vert _{\underline{L}^{\frac{2d}{d+2}}(B_r(z))}+\Vert g\Vert _{\underline{L}^2(B_r(z))}. \end{aligned}$$

(154)

Next, we suppose that $B_r(z)\setminus B\ne \emptyset$. We easily obtain the following Caccioppoli inequality

$$\begin{aligned} \Vert \nabla u\Vert _{\underline{L}^2(B_\frac{r}{2}(z))}\lesssim r^{-1}\Vert u\Vert _{\underline{L}^2(B_{r}(z))}+\Vert g\Vert _{\underline{L}^2(B_{r}(z))}. \end{aligned}$$

Clearly, we find $c=c(d)>0$ such that

$$\begin{aligned} |B_{2r}(z)\setminus B|\ge c|B_{2r}(z)|. \end{aligned}$$

Hence, we can apply a version of Poincaré inequality (see e.g., [20, Theorem 3.16]) to obtain

$$\begin{aligned} \Vert \nabla u\Vert _{\underline{L}^2(B_\frac{r}{2}(z))}\lesssim&r^{-1}\Vert u\Vert _{\underline{L}^2(B_{2r}(z))}+\Vert g\Vert _{\underline{L}^2(B_{2r}(z))}\nonumber \\ \lesssim&\Vert \nabla u\Vert _{\underline{L}^{\frac{2d}{d+2}}(B_{2r}(z)}+\Vert g\Vert _{\underline{L}^2(B_{2r}(z)}. \end{aligned}$$

(155)

Clearly, (154), (155) and a simple covering argument imply (153) which finishes the proof. $\square$