We shall now address the question of existence of solutions to the problem (8.11b), (8.5b).

Theorem 9.1

Fix with , \(M\ge 0\), and let \(f\in \mathcal C(M)\). There exists a small constant \(R>0\) depending only on and M such that if

then Eq. (8.11b) has a solution \(\phi ^\mu \in C^2([0,1])\) satisfying

$$\begin \phi ^\mu (0)=D^2_r\phi ^\mu (0)=0,\quad D_r\phi ^\mu (0)=1, \end$$

(9.1a)

as well as the estimates

$$\begin |\phi ^\mu (r)/r-1|,\;|D_r\phi ^\mu (r)-1|,\;|u^\mu (r)-1| \le Rr^2, \end$$

(9.1b)

and

$$\begin |D_r^2\phi ^\mu (r)|\le Rr, \end$$

(9.1c)

for \(r\in [0,1]\).

The map from to C([0, 1]) given by

$$\begin \mu \mapsto \phi ^\mu \end$$

is continuous.

If \(\mu >0\), then

$$\begin \lambda _1^\mu (r)=D_r\phi ^\mu (r)>\lambda _2^\mu (r)=\phi ^\mu (r)/r>1,\quad r\in (0,1], \end$$

(9.2a)

and if \(\mu <0\), then

$$\begin \lambda _1^\mu (r)<\lambda _2^\mu (r)<1,\quad r\in (0,1]. \end$$

(9.2b)

If \(\beta (f)\) is sufficiently large, then there exists an eigenvalue \(\mu \ne 0\) with such that the solution \(\phi ^\mu \) satisfies the boundary condition (8.5b).

Remark

The assumption that \(D_r\phi ^\mu (0)=1\) in (9.1a) does not restrict the possible initial data of the motion in (3.2).

Proof of Theorem 9.1

In order to handle the apparent singularity at \(r=0\), it is convenient to make the ansatz

$$\begin \phi (r)=r+K\zeta (r)\equiv r+\int _0^r(r-\rho )\rho \zeta (\rho )d\rho ,\quad \text \quad \zeta \in C([0,1]). \end$$

(9.3)

Notice that K is a bounded linear operator from C([0, 1]) into \(C^2([0,1])\). In fact, it follows from (9.3) that

$$\begin \begin&\phi '(r)=\lambda _1(r)=1+\int _0^r\rho \zeta (\rho )d\rho \\&\phi (r)/r=\lambda _2(r)=1+\frac\int _0^r(r-\rho )\rho \zeta (\rho )d\rho \\&\phi '(r)-\phi (r)/r=\lambda _1(r)-\lambda _2(r) =\frac\int _0^r\rho ^2\zeta (\rho )d\rho \\&\phi ''(r)=r\zeta (r). \end \end$$

(9.4a)

In particular, \(\phi (r)=r+K\zeta (r)\) satisfies the conditions (9.1a).

Assume now that (9.3) holds with

$$\begin \zeta \in N_R=\,\quad R\le \delta , \end$$

where was previously defined in (8.9).

Straightforward pointwise estimates for \(r\in [0,1]\) yield

$$\begin \begin&|\phi (r)-r|\le \tfrac\Vert \zeta \Vert _\infty r^3\le \tfrac Rr^3\\&|\phi '(r)-1|=|\lambda _1(r)-1|<}\Vert \zeta \Vert _\infty r^2\le }Rr^2\\&|\phi (r)/r-1|=|\lambda _2(r)-1|\le \tfrac\Vert \zeta \Vert _\infty r^2\le \tfrac Rr^2\\&|\lambda _1(r)-\lambda _2(r)|\le \tfrac\Vert \zeta \Vert _\infty r^2\le \tfracRr^2\\&\lambda _2(r)\ge 1-|\lambda _2(r)-1|\ge 1-\tfracRr^2\ge 2/3\\&|u(r)-1|=\left| \lambda _2(r)^(\lambda _1(r) -\lambda _2(r))\right| \le \tfrac\Vert \zeta \Vert _\infty r^2\le }Rr^2\\&|v(r)-1|=\left| \lambda _1(r)\lambda _2(r)^2 -1\right| \le 2\Vert \zeta \Vert _\infty r^2\le 2Rr^2. \end \end$$

(9.4b)

By (9.4b), it follows that (9.1b), (9.1c) hold, and as a consequence (5.1), (8.11a) also are valid.

Since \(\zeta \in N_R\) is small, we regard \(\phi (r)=r+K\zeta (r)\) as a perturbation of the identity map. Explicitly, \(\zeta (s)\equiv 0\) implies that

$$\begin \phi (r)=r\quad \text \quad \lambda _1(r)=\lambda _2(r)=u(r)=v(r)=1. \end$$

Note that \(\phi (r)=r\) solves (8.11b) with \(\mu =0\).

Making the substitution (9.3) in equation (8.11b) and using (9.4a), we find that

$$\begin L\zeta (r)=\mathcal F(\zeta ,\mu )(r), \end$$

(9.5)

with

$$\begin L\zeta (r)= \zeta (r)+\frac\int _0^r\rho ^2\zeta (\rho )d\rho \end$$

and

Recall that the functions \(V_i(u)\) depend on \(f\in \mathcal C(M)\) and satisfy the conditions (8.10b), (8.10c).

The operator L is an isomorphism on C([0, 1]) with bounded inverse

$$\begin L^\eta (r)=\eta (r)-\frac\int _0^r\rho ^4\eta (\rho )d\rho . \end$$

From (9.5), we arrive at the reformulation

$$\begin \zeta (r)=L^\mathcal F(\zeta ,\mu )(r). \end$$

(9.6)

In order to solve (8.11b), (9.1a), it is sufficient to find a solution \(\zeta \) of (9.6) in \(N_R\), with \(R\le \delta \).

Let us define

since this expression appears repeatedly.

The claim is that for \(|\varepsilon |\le R\ll 1\) the map

$$\begin \zeta \mapsto L^\mathcal F(\zeta ,\mu )\end$$

is a uniform contraction on \(N_R\) taking \(N_R\) into itself.

Assume that

$$\begin |\varepsilon |\le R\le \delta . \end$$

(9.7)

As a consequence of (8.10b), (8.10c), (9.4b), there exists a constant \(C_1\), independent of R and \(\beta (f)\), such that

$$\begin & \Vert \mathcal F(\zeta _1,\mu )-\mathcal F(\zeta _2,\mu )\Vert _\infty \nonumber \\ & \quad \le C_1 \left( R+\varepsilon \right) \Vert \zeta _1-\zeta _2\Vert _\infty \le 2C_1 R\Vert \zeta _1-\zeta _2\Vert _\infty , \end$$

(9.8a)

for all \(\zeta _1,\zeta _2\in N_R\). By (8.10b), we have \(\mathcal F(0,\mu )(r)=\varepsilon \). It follows from (9.8a) that

$$\begin \Vert \mathcal F(\zeta ,\mu )-\varepsilon \Vert _\infty \le 2C_1R^2, \end$$

(9.8b)

for all \(\zeta \in N_R\).

For the contraction estimate, we have from (9.8a)

$$\begin & \Vert L^\mathcal F(\zeta _1,\mu )-L^\mathcal F(\zeta _2,\mu )\Vert _\infty \nonumber \\ & \quad \le \Vert L^\Vert \;2C_1R\;\Vert \zeta _1-\zeta _2\Vert _\infty \le (1/10)\Vert \zeta _1-\zeta _2\Vert _\infty , \end$$

(9.9a)

for all \(\zeta _1,\zeta _2\in N_R\), provided \(\Vert L^\Vert 2C_1R\le 1/10\).

Since \(L^\varepsilon =3\varepsilon /5\) for any constant function \(\varepsilon \), by (9.8b), (9.7), there holds

$$\begin \Vert L^\mathcal F(\zeta ,\mu )\Vert _\infty= & \Vert L^(\mathcal F(\zeta ,\mu )-\varepsilon )+3\varepsilon /5\Vert _\infty \nonumber \\ \le & \Vert L^\Vert 2C_1R^2+3\varepsilon /5\le R/10+3R/5 < R,\nonumber \\ \end$$

(9.9b)

for all \(\zeta \in N_R\), provided \(\Vert L^\Vert 2C_1R\le 1/10\). This shows that the map leaves \(N_R\) invariant.

This establishes the claim under the restrictions

$$\begin R\le \delta \quad \text \quad 2C_1\Vert L^\Vert R\le 1/10. \end$$

(9.10)

We emphasize that these restrictions on R do not depend on the value of \(\beta (f)\), and therefore, the estimates (9.9a), (9.9b) hold for all \(f\in \mathcal C(M)\).

By the uniform contraction principle (see [5] Section 2.2, or [23] Section 5.3), the equation (9.6) has a unique solution \(\zeta ^\mu \in N_R\subset C([0,1])\), for each such that \(|\varepsilon |\le R\). Moreover, the map \(\mu \mapsto \zeta ^\mu \), from \(\\) to C([0, 1]), is continuous. In particular, \(\zeta ^0(r)\equiv 0\), by uniqueness.

For each such \(\zeta ^\mu \), we obtain a \(C^2\) solution

of (8.11b) which satisfies (9.1a), (9.1b), (9.1c) and depends continuously on the parameter \(\mu \).Footnote 2

Since, by definition (8.4c),

$$\begin D_r\lambda _2^\mu (r)=\frac\left( \lambda _1^\mu (r)-\lambda _2^\mu (r)\right) , \end$$

(9.11)

it follows from (8.11b) that

This, in turn, may be expressed in the form

$$\begin D_r\mathcal X^\mu (r)=\mathcal Y^\mu (r)\mathcal X^\mu (r)+\mu \mathcal Z^\mu (r), \end$$

where

Note that, by (8.10b), (9.4b), the coefficients \(\mathcal Y^\mu \), \(\mathcal Z^\mu \) are continuous on [0, 1] and \(\mathcal Z^\mu \) is strictly positive on (0, 1], by (8.10b). So we can write

$$\begin \mathcal X^\mu (r)=\mu \int _0^r\left( \exp \int _0^\rho \mathcal Y^\mu (s)ds\right) \mathcal Z^\mu (\rho )d\rho . \end$$

We conclude that if \(\mu >0\), then \(\mathcal X^\mu \) is strictly positive on (0, 1], and then from (9.11), that \(D_r\lambda _2^\mu (r)\) is strictly positive on (0, 1]. Thus, from (9.1a), we have

$$\begin \lambda _1^\mu (r)>\lambda _2^\mu (r)>\lambda _2^\mu (0)=D_r\phi ^\mu (0)=1, \quad r\in (0,1]. \end$$

This proves (9.2a), and (9.2b) follows analogously.

For future reference, we also record the fact that

$$\begin \,}}(u^\mu (1)-1)=\,}}\mathcal X^\mu (1)=\,}}\mu . \end$$

(9.12)

We shall now show that for all with and \(\beta (f)\) sufficiently large, the eigenvalue \(\mu \) may be chosen so that the boundary condition (8.5b) is fulfilled.

We continue to assume that R satisfies the conditions (9.10). Define

and consider the solution family

$$\begin \(r)=r+K\zeta ^(r):\}. \end$$

The first step will be to establish lower bounds for \(|u^(1)-1|\). Since \(\zeta ^\) solves (9.6), we have from (9.8b), (9.10),

$$\begin \Vert \zeta ^-3\varepsilon /5\Vert _\infty= & \Vert L^(\mathcal F(\zeta ^,\mu (\varepsilon ))-\varepsilon )\Vert _\infty \\\le & \Vert L^\Vert 2C_1R^2\le R/10. \end$$

Letting \(\varepsilon =\pm R\), this implies that

$$\begin \zeta ^(r)\ge R/2\quad \text \quad \zeta ^(r)\le -R/2,\quad \text \quad r\in [0,1]. \end$$

(9.13)

From (9.4a), (9.4b), (9.13), we obtain

$$\begin \begin&\lambda _1^(1)-\lambda _2^(1) =\int _0^1\rho ^2\zeta ^(\rho )d\rho \ge (R/2)(1/3)=R/6\\&\lambda _1^(1)-\lambda _2^(1) =\int _0^1\rho ^2\zeta ^(\rho )d\rho \le -R/6\\&1<\lambda _2^(1)=1+\int _0^1(1-\rho )\rho \zeta ^(\rho )d\rho \le 1+R/6<7/6\\&1>\lambda _2^(1)=1+\int _0^1(1-\rho )\rho \zeta ^(\rho )d\rho \ge 1-R/6>5/6. \end \end$$

These estimates combine to show that

$$\begin \begin&u^(1)-1 =\frac(1)-\lambda _2^(1)}(1)} \ge \frac=R/7\\&u^(1)-1 =\frac(1)-\lambda _2^(1)}(1)} \le \frac<-R/7. \end \end$$

(9.14)

Suppose now that \(f\in \mathcal C(M)\) with \(\beta (f)\) sufficiently large:

(9.15)

Then

so that by Lemma 8.3, g has a unique zero . By continuous dependence upon parameters, the function

$$\begin z(\varepsilon )=u^(1)-1 \end$$

is continuous for \(|\varepsilon |\le R\). Using (9.14), (9.15), we find that

So there exists \(|\varepsilon _0|<R\) such that \(z(\varepsilon _0)=u_0-1\), and thus,

$$\begin g(u^(1))=g(u_0)=0. \end$$

By Lemma 8.3 and (9.12), we also have that

Therefore, we have shown that if the eigenvalue is taken to be , then \(\phi ^\) satisfies the boundary condition (8.5b) and . \(\square \)

Remark

The boundary condition \(u^(1)=u_0\) is equivalent to the linear and homogeneous Robin boundary condition

$$\begin D_r\phi ^(1)=u_0\phi ^(1). \end$$

Corollary 9.2

Fix with and \(M\ge 0\). Suppose that W satisfies (2.1a), (2.1b), (2.1c), (2.1d), and using Lemma 8.1,

for all \(\lambda \in \mathbb _+^2\) and \(\omega \in S^2\). Let S be defined by (2.4a).

If \(\beta (f)\) is sufficiently large, then there exists a \(C^2\) spherically symmetric orientation-preserving deformation \(\varphi :\overline\rightarrow \mathbb ^3\) and an eigenvalue \(\mu \) with satisfying (3.4a), (3.4b).

If , then \(\varphi (\overline)\supset \overline\) and the principal stretches satisfy \(\lambda _1(r)>\lambda _2(r)>1\), \(r\in (0,1]\), while when , the inclusion and inequalities are reversed.

Proof

If \(\beta (f)\) is sufficiently large, Theorem 9.1 ensures the existence of an eigenvalue \(\mu \in \mathbb \), with , and an eigenfunction function \(\phi =\phi ^\mu \in C^2([0,1])\) satisfying the hypotheses of Lemma 8.3, the differential equation (8.11b), and the boundary condition (8.5b). Therefore, \(\phi \) also solves the differential equation (8.5a).

By assumption, f(u) is the restriction of an appropriate strain energy function to the spherically symmetric deformation gradients. Therefore, Lemma 8.2 yields the desired deformation \(\varphi (y)=\phi (r)\omega \).

Since \(\varphi (\overline)\) is a sphere of radius \(\phi (1)=\lambda _2(1)\), the final statements follow from (9.2a), (9.2b). \(\square \)

Remark

Since we have assumed that the reference density is \(\bar=1\), the density of the deformed configuration in material coordinates is \(\varrho \circ \varphi (y)=v(r)^=\left( \lambda _1(r)\lambda _2(r)^2\right) ^\). If , we have \(\varrho \circ \varphi (y)<1\), for \(r\in (0,1]\), and if , we have \(\varrho \circ \varphi (y)>1\), for \(r\in (0,1]\).