On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

Abstract

We study the asymptotic behavior of solutions with finite energy to the displacement problem of linear elastostatics in a three-dimensional exterior Lipschitz domain.

1. Introduction

The displacement problem of elastostatics in an exterior Lipschitz domain $\Omega$ of ${\mathbb{R}}^3$ consists of finding a solution to the equations [1] (Notation—Unless otherwise specified, we will use the notation of the classical monograph [1] by M.E. Gurtin. In particular, ${\left(\mathrm{div}\mathbf{C}[\nabla \mathit{u}]\right)}_i={\partial }_j\left({\mathrm{C}}_{ijhk}{\partial }_k{u}_h\right)$, Lin is the space of second–order tensors (linear maps from ${\mathbb{R}}^3$ into itself) and Sym, Skw are the spaces of the symmetric and skew elements of Lin respectively; if $\mathit{E}\in \mathrm{Lin}$ and $\mathit{v}\in R^3$, $\mathit{E}\mathit{v}$ is the vector with components $E_{ij}{v}_j$. $B_R=\{x\in {\mathbb{R}}^3:r=\left|\mathit{x}\right|<R\}$, $T_R=B_{2R}\setminus B_R$, $\complement B_R=R^3\setminus \overline{B_R}$ and $B_{R_0}$ is a fixed ball containing $\partial \Omega$. If $f\left(x\right)$ and $\varphi \left(r\right)$ are functions defined in a neighborhood of infinity, then $f\left(x\right)=o\left(\varphi \right(r\left)\right)$ means that ${lim}_{r\to +\infty }(f/\varphi )=0$. To lighten up the notation, we do not distinguish between scalar, vector, and second–order tensor space functions; c will denote a positive constant whose numerical value is not essential to our purposes.)

$$\begin{array}{cc}\hfill \mathrm{div}\mathbf{C}[\nabla \mathit{u}]& =\mathit{0}\mathrm{in}\Omega ,\hfill \\ \hfill \mathit{u}& =\widehat{\mathit{u}}\mathrm{on}\partial \Omega ,\hfill \\ \hfill {\displaystyle \underset{R\to +\infty }{lim}\underset{\partial B}{\int }\mathit{u}(R,\sigma )d\sigma }& =\mathit{0},\hfill \end{array}$$

(1)

where $\mathit{u}$ is the (unknown) displacement field, $\widehat{\mathit{u}}$ is an (assigned) boundary displacement, B is the unit ball, $\mathbf{C}\equiv \left[{\mathrm{C}}_{ijhk}\right]$ is the (assigned) elasticity tensor, i.e., a map from $\Omega \times \mathrm{Lin}\to \mathrm{Sym}$, linear on Sym and vanishing in $\Omega \times \mathrm{Skw}$. We shall assume $\mathbf{C}$ to be symmetric, i.e.,

$$\mathit{E}·\mathbf{C}\left[\mathit{L}\right]=\mathit{L}·\mathbf{C}\left[\mathit{E}\right]\forall \mathit{E},\mathit{L}\in \mathrm{Lin},$$

(2)

and positive definite, i.e., there exists positive scalars ${\mu }_0$ and ${\mu }_e$ (minimum and maximum elastic moduli [1]) such that

$${\mu }_0{\left|\mathit{E}\right|}^2\le \mathit{E}·\mathbf{C}\left[\mathit{E}\right]\le {\mu }_e{\left|\mathit{E}\right|}^2,\forall \mathit{E}\in \mathrm{Sym},\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega .$$

(3)

Let $D^{1,q}(\Omega )$, $D_0^{1,q}(\Omega )$$(q\in [1,+\infty \left)\right)$ be the completion of $C_0^{\infty }\left(\overline{\Omega }\right)$ and $C_0^{\infty }(\Omega )$, respectively, with respect to the norm ${\parallel \nabla \mathit{u}\parallel }_{L^q(\Omega )}$.

We consider solutions $\mathit{u}$ to equations (1) with finite Dirichlet integral (or with finite energy) that we call D-solutions analogous with the terminology used in viscous fluid dynamics (see [2]). More precisely, we say that $\mathit{u}\in D^{1,2}(\Omega )$ is a D-solution to equation (1)${}_1$

$$\underset{\Omega }{\int }\nabla \phi ·\mathbf{C}[\nabla \mathit{u}]=0,\forall \phi \in D_0^{1,2}(\Omega ).$$

(4)

A D-solution to system (1) is a D-solution to equation (1)${}_1$, which satisfies the boundary condition in the sense of trace in Sobolev’s spaces and tends to zero at infinity in a mean square sense [2]

$$\underset{R\to +\infty }{lim}\underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{2}d\sigma =0.$$

(5)

If $\mathit{u}$ is a D-solution to (1)${}_1$, then the traction field on the boundary is

$$\mathit{s}\left(\mathit{u}\right)=\mathbf{C}[\nabla \mathit{u}]\mathit{n}$$

where a well defined field of $W^{-1/2,2}(\partial \Omega )$ exists and the following generalized work and energy relation [1] holds

$$\underset{\Omega \cap B_R}{\int }\nabla \mathit{u}·\mathbf{C}[\nabla \mathit{u}]=\underset{\partial \Omega }{\int }\mathit{u}·\mathit{s}\left(\mathit{u}\right),$$

where abuse of notation ${\int }_{\partial \Omega }\mathit{u}·\mathit{s}\left(\mathit{u}\right)$ means the value of the functional $\mathit{s}\left(\mathit{u}\right)\in W^{-1/2,2}(\partial \Omega )$ at $\mathit{u}\in W^{1/2,2}(\partial \Omega )$, and $\mathit{n}$ is the unit outward (with respect to $\Omega$) normal to $\partial \Omega$.

If $\widehat{\mathit{u}}\in W^{1/2,2}(\partial \Omega )$, denoting by ${\mathit{u}}_0\in D^{1,2}(\Omega )$ an extension of $\widehat{\mathit{u}}$ in $\Omega$ vanishing outside a ball, then ${\left(1\right)}_{1,2}$ is equivalent to finding a field $\mathit{u}\in D_0^{1,2}(\Omega )$ such that

$$\underset{\Omega }{\int }\nabla \phi ·\mathbf{C}[\nabla \mathit{v}]=-\underset{\Omega }{\int }\nabla \phi ·\mathbf{C}[\nabla {\mathit{u}}_0],\forall \phi \in D_0^{1,2}(\Omega ).$$

(6)

Since the right-hand side of (6) defines a linear and continuous functional on $D_0^{1,2}(\Omega )$, and by the first Korn inequality (see [1] Section 13)

$$\underset{\Omega }{\int }{|\nabla \mathit{v}|}^2\le \frac{2}{{\mu }_0}\underset{\Omega }{\int }\nabla \mathit{v}·\mathbf{C}[\nabla \mathit{v}],$$

by the Lax–Milgram lemma, (6) has a unique solution $\mathit{v}$, and the field $\mathit{u}=\mathit{v}+{\mathit{u}}_0$ is a D-solution to ${\left(1\right)}_{1,2}$. It satisfies ${\left(1\right)}_3$ in the following sense (see Lemma 1)

$$\underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{2}d\sigma =o\left(R^{-1}\right).$$

(7)

Moreover, $\mathit{u}$ exhibits more regularity properties provided $\mathbf{C}$, $\partial \Omega$ and $\widehat{\mathit{u}}$ are more regular. In particular, if $\mathbf{C}$, $\widehat{\mathit{u}}$ and $\partial \Omega$ are of class $C^{\infty }$, then $\mathit{u}\in C^{\infty }\left(\overline{\Omega }\right)$ [3].

If $\mathbf{C}$ is constant, then existence and regularity hold under the weak assumption of strong ellipticity [1], i.e.,

$${\mu }_0{\left|\mathit{a}\right|}^2{\left|\mathit{b}\right|}^2\le a·\mathbf{C}[\mathit{a}\otimes \mathit{b}]\mathit{b},\forall \mathit{a},\mathit{b}\in R^3.$$

(8)

As far as we are aware, except for the property (7), little is known about the convergence at infinity of a D-solution and, in particular, whether or under what additional conditions (7) can be improved.

The main purpose of this paper is just to determine reasonable conditions on $\mathbf{C}$ assuring that (7) can be improved.

We say that $\mathbf{C}$ is regular at infinity if there is a constant elasticity tensor ${\mathbf{C}}_0$ such that

$$\underset{\left|x\right|\to +\infty }{lim}\mathbf{C}\left(x\right)={\mathbf{C}}_0.$$

(9)

Let ${\mathfrak{C}}_0$ and $\mathfrak{C}$ denote the linear spaces of the D-solutions to the equations

$$\begin{array}{cc}\hfill \mathrm{div}\mathbf{C}[\nabla \mathit{h}]& =\mathit{0}\mathrm{in}\Omega ,\hfill \\ \hfill \mathit{h}& =\tau \mathrm{on}\partial \Omega ,\hfill \\ \hfill {\displaystyle \underset{R\to +\infty }{lim}\underset{\partial B}{\int }{\left|\mathit{h}(R,\sigma )\right|}^{2}d\sigma }& =\mathit{0},\hfill \end{array}$$

(10)

for all $\tau \in R^3$ and

$$\begin{array}{cc}\hfill \mathrm{div}\mathbf{C}[\nabla \mathit{h}]& =\mathit{0}\mathrm{in}\Omega ,\hfill \\ \hfill \mathit{h}& =\tau +\mathit{A}\mathit{x}\mathrm{on}\partial \Omega ,\hfill \\ \hfill {\displaystyle \underset{R\to +\infty }{lim}\underset{\partial B}{\int }{\left|\mathit{h}(R,\sigma )\right|}^{2}d\sigma }& =\mathit{0},\hfill \end{array}$$

(11)

for all $\tau \in R^3$, $\mathit{A}\in \mathrm{Lin}$, respectively.

The following theorem holds.

Theorem 1.

Let $\mathit{u}$ be the D–solution to $\left(1\right)$. There is $q<2$ depending only on $\mathbf{C}$ such that

$$\underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{q}d\sigma =o\left(R^{q-3}\right).$$

(12)

If $\mathbf{C}$ is regular at infinity, then

$$\underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{q}d\sigma =o\left(R^{q-3}\right),\forall q\in (3/2,+\infty ),$$

(13)

and

$$\underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{q}d\sigma =o\left(R^{q-3}\right),\forall q\in (1,2]⟺\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right)=\mathit{0},\forall \mathit{h}\in {\mathfrak{C}}_0.$$

(14)

Moreover, if

$$\underset{\partial \Omega }{\int }\mathbf{C}[\widehat{\mathit{u}}\otimes \mathit{n}]=\mathit{0},\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right)=\mathit{0},\forall \mathit{h}\in \mathfrak{C},$$

(15)

then

$$\underset{\partial B}{\int }\left|\mathit{u}(R,\sigma )\right|d\sigma =o\left(R^{-2}\right).$$

(16)

2. Preliminary Results

In this section, we collect the main tools we need to prove Theorem 1.

Lemma 1.

If $u\in D^{1,q}(\Omega )$, for $q\in [1,2]$, then

$$\underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{q}d\sigma \le c\left(q\right)R^{q-3}\underset{\complement B_R}{\int }{|\nabla \mathit{u}|}^q.$$

(17)

Moreover, if $q=2$, then, for $R\gg R_0$,

$$\underset{\complement B_R}{\int }\frac{{\left|\mathit{u}\right|}^2}{r^2}\le 4\underset{\complement B_R}{\int }{|\nabla \mathit{u}|}^2.$$

(18)

Proof. 

Lemma 1 is well-known (see, e.g., [2,4] and [5] Chapter II). We propose a simple proof for the sake of completeness. Since $D^{1,2}(\Omega )$ is the completion of $C_0^{\infty }\left(\overline{\Omega }\right)$ with respect to the norm ${\parallel \nabla \mathit{u}\parallel }_{L^2(\Omega )}$, it is sufficient to prove (17) and (18) for a regular field $\mathit{u}$ vanishing outside a ball. By basic calculus and Hölder inequality,

$$\begin{array}{cc}\hfill {\displaystyle \underset{\partial B}{\int }{\left|\mathit{u}(R,\sigma )\right|}^{q}d\sigma }& {\displaystyle =\underset{\partial B}{\int }|\underset{R}{\overset{+\infty }{\int }}{\partial }_r\mathit{u}(r,\sigma )dr{|}^{q}d\sigma =\underset{\partial B}{\int }|\underset{R}{\overset{+\infty }{\int }}{r}^{2/q}{r}^{-2/q}{\partial }_r\mathit{u}(r,\sigma )dr{|}^{q}d\sigma }\hfill \\ & {\displaystyle \le \left\{\underset{\partial B}{\int }d\sigma \underset{R}{\overset{+\infty }{\int }}{|\nabla \mathit{u}(r,\sigma )|}^q{r}^{2}dr\right\}\left\{\underset{\partial B}{\int }|\underset{R}{\overset{+\infty }{\int }}{r}^{-\frac{2}{q-1}}dr{|}^{q-1}d\sigma \right\}.}\hfill \end{array}$$

Hence, (17) follows by a simple integration.

From

$$\underset{\complement B_R}{\int }\frac{{\left|\mathit{u}\right|}^2}{r^2}=\underset{R}{\overset{+\infty }{\int }}{\partial }_r\left(r\underset{\partial B}{\int }{\left|\mathit{u}(r,\sigma )\right|}^{2}d\sigma \right)-2\underset{\complement B_R}{\int }\frac{\mathit{u}}{r}·{\partial }_r\mathit{u}$$

by Schwarz’s inequality, one gets

$$\underset{\complement B_R}{\int }\frac{{\left|\mathit{u}\right|}^2}{r^2}\le 2{\left\{\underset{\complement B_R}{\int }\frac{{\left|\mathit{u}\right|}^2}{r^2}\underset{\complement B_R}{\int }{|\nabla \mathit{u}|}^2\right\}}^{1/2}.$$

Hence, (18) follows at once. ☐

Let ${\mathbf{C}}_0$ be a constant and strongly elliptic elasticity tensor. The equation

$$\mathrm{div}{\mathbf{C}}_0[\nabla \mathit{u}]=\mathit{0}$$

(19)

admits a fundamental solution $\mathcal{U}(x-y)$ [6] that enjoys the same qualitative properties as the well-known ones of homogeneous and isotropic elastostatics, defined by

$${\mathcal{U}}_{ij}(x-y)=\frac{1}{8\pi \mu (1-\nu )|x-y|}\left[(3-4\nu ){\delta }_{ij}+\frac{(x_i-y_i)(x_j-z_j)}{{|x-y|}^2}\right],$$

where $\mu$ is the shear modulus and $\nu$ the Poisson ratio (see [1] Section 51). In particular, $\mathcal{U}\left(x\right)=O\left(r^{-1}\right)$ and for $\mathit{f}$ with compact support (say) the volume potential

$$\mathcal{V}\left[\mathit{f}\right]\left(x\right)={\int }_{R^3}\mathcal{U}(x-y)\mathit{f}\left(y\right)d{v}_y$$

(20)

is a solution (in a sense depending on the regularity of $\mathit{f}$) to the system

$$\mathrm{div}{\mathbf{C}}_0[\nabla \mathit{u}]+\mathit{f}=\mathit{0}\mathrm{in}{R}^3.$$

(21)

Let ${\mathcal{H}}^1$ denote the Hardy space on $R^3$ (see [7] Chapter III). The following result is classical (see, e.g., [7]).

Lemma 2.

${\nabla }_2\mathcal{V}$ maps boundedly $L^q$ into itself for $q\in (1,+\infty )$ and ${\mathcal{H}}^1$ into itself.

Lemma 3.

Let u be the D-solution to $\left(1\right)$, Then, for $R\gg R_0$,

$$\underset{\partial \Omega }{\int }\mathit{s}\left(\mathit{u}\right)=\underset{\partial B_R}{\int }\mathit{s}\left(\mathit{u}\right),$$

(22)

and

$$\underset{\partial \Omega }{\int }\mathit{h}·\mathit{s}\left(\mathit{u}\right)=\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right),\forall \mathit{h}\in \mathfrak{C},$$

(23)

where $\mathfrak{C}$ is the space of D-solutions to system $\left(11\right)$.

Proof. 

Let

$$g\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \left|x\right|>2R,\hfill \\ 1,\hfill & \left|x\right|<R,\hfill \\ R^{-1}(R-|x\left|\right),\hfill & R\le \left|x\right|\le 2R,\hfill \end{array}\right.$$

(24)

with $R\gg R_0$. Scalar multiplication of both sides of (1)${}_1$ by $g\mathit{h}$, (2) and an integration by parts yield

$$\underset{\partial \Omega }{\int }\mathit{h}·\mathit{s}\left(\mathit{u}\right)-\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right)=\frac{1}{R}\underset{T_R}{\int }\left(\mathit{u}·\mathbf{C}[\nabla \mathit{h}]{\mathit{e}}_r-\mathit{h}·\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_r\right),$$

where ${\mathit{e}}_r=\mathit{x}/r$. Since $R\le \left|x\right|\le 2R$, by Schwarz’s inequality,

$$\begin{array}{c}\hfill {\displaystyle |\frac{1}{R}\underset{T_R}{\int }\mathit{h}·\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_r|\le 2\underset{T_R}{\int }{r}^{-1}|\mathit{h}·\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_r|\le c\parallel r^{-1}{\mathit{h}\parallel }_{L^2\left(T_R\right)}{\parallel \nabla \mathit{u}\parallel }_{L^2\left(T_R\right)},}\\ \hfill {\displaystyle |\frac{1}{R}\underset{T_R}{\int }\mathit{u}·\mathbf{C}[\nabla \mathit{h}]{\mathit{e}}_r|\le 2\underset{T_R}{\int }{r}^{-1}|\mathit{u}·\mathbf{C}[\nabla \mathit{h}]{\mathit{e}}_r|\le c\parallel r^{-1}{\mathit{u}\parallel }_{L^2\left(T_R\right)}{\parallel \nabla \mathit{h}\parallel }_{L^2\left(T_R\right)}.}\end{array}$$

Hence, letting $R\to +\infty$ and taking into account Lemma 1, (23) follows. ☐

Lemma 4.

Let $\mathit{u}$ be the D-solution to $\left(1\right)$; then, for $R\gg R_0$,

$$\underset{\partial \Omega }{\int }\left(\mathit{x}\otimes \mathit{s}\left(\mathit{u}\right)-\mathbf{C}[\widehat{\mathit{u}}\otimes \mathit{n}]\right)=\underset{\partial B_R}{\int }\left(\mathit{x}\otimes \mathit{s}\left(\mathit{u}\right)-\mathbf{C}[\mathit{u}\otimes {\mathit{e}}_R]\right),$$

(25)

where $\mathfrak{C}$ is the space of D-solutions to system $\left(11\right)$.

Proof. 

(25) is easily obtained by integrating the identity

$$\mathit{0}=\mathit{x}\otimes \mathrm{div}\mathbf{C}[\nabla \mathit{u}]=\mathrm{div}\left(\mathit{x}\otimes \mathbf{C}[\nabla \mathit{u}]\right)-\mathbf{C}[\nabla \mathit{u}]$$

over $B_R$ and using the divergence theorem. ☐

3. Proof of Theorem 1

Let $\vartheta \left(r\right)$ be a regular function, vanishing in $B_R$ and equal to 1 outside $B_{2R}$, for $R\gg R_0$. The field $v=\vartheta \mathit{u}$ is a D-solution to the equation

$$\mathrm{div}\mathbf{C}[\nabla \mathit{v}]+\mathit{f}=\mathit{0}\mathrm{in}{R}^3,$$

(26)

with

$$\mathit{f}=-\mathbf{C}[\nabla \mathit{u}]\nabla \vartheta -\mathrm{div}\mathbf{C}[\mathit{u}\otimes \nabla \vartheta ].$$

(27)

Of course, $\mathit{f}\in L^2\left(R^3\right)$ vanishes outside $T_R$. Let ${\mathbf{C}}_0$ be a strongly elliptic elasticity tensor. Clearly, $\mathit{v}$ is a D-solution to the system

$$\mathrm{div}{\mathbf{C}}_0[\nabla \mathit{v}]+\mathrm{div}(\mathbf{C}-{\mathbf{C}}_0)[\nabla \mathit{v}]+\mathit{f}=\mathit{0}\mathrm{in}{R}^3,$$

(28)

which coincides with $\mathit{u}$ outside $B_{2R}$. Since

$${\nabla }_k\mathcal{V}\left[\mathit{f}\right]\left(x\right)=O\left(r^{-1-k}\right),k\in N,{\nabla }_k=\underset{k-\mathrm{times}}{\underbrace{\nabla \dots \nabla }},$$

(29)

by Lemma 2, the map

$${\mathit{w}}^{\prime }\left(x\right)=\nabla \mathcal{V}\left[(\mathbf{C}-{\mathbf{C}}_0)[\nabla \mathit{w}]\right]\left(x\right)+\mathcal{V}\left[\mathit{f}\right]\left(x\right)$$

(30)

is continuous from $D^{1,q}$ into itself, for $q\in (3/2,+\infty )$. Choose

$${{\mathbf{C}}_0}_{ijhk}={\mu }_e{\delta }_{ih}{\delta }_{jk}.$$

Since

$$\parallel \nabla \mathcal{V}\left[(\mathbf{C}-{\mathbf{C}}_0)[\nabla \mathit{w}]\right]{\parallel }_{D^{1,q}}\le c\left(q\right)\frac{{\mu }_e-{\mu }_0}{{\mu }_e}{\parallel \mathit{w}\parallel }_{D^{1,q}}$$

and [7]

$$\underset{q\to 2}{lim}c\left(q\right)=1,$$

the map (30) is contractive in a neighborhood of 2 and its fixed point must coincide with $\mathit{v}$ . Hence, there is $q\in (1,2)$ such that $\mathit{u}\in D^{1,q}(\Omega )$ and (12) is proved.

If $\mathbf{C}$ is regular at infinity, then by Lemma 1 and the property of $\vartheta$,

$$\parallel {\mathit{v}}^{\prime }-{\mathit{z}}^{\prime }{\parallel }_{D^{1,q}}\le c\left(q\right)\parallel \mathbf{C}-{\mathbf{C}}_0{\parallel }_{L^{\infty }\left(\complement S_{R_0}\right)}{\parallel \mathit{v}-\mathit{z}\parallel }_{D^{1,q}}.$$

(31)

Since the constant $c\left(q\right)$ is uniformly bounded in every interval $[a,b]$ and $\parallel \mathbf{C}-{\mathbf{C}}_0{\parallel }_{L^{\infty }\left(\complement S_{R_0}\right)}$ is sufficiently small, $u\in D^{1,q}$ for $\mathit{q}\in (3/2,+\infty )$.

Assume that

$$\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right)=\mathit{0},\forall \mathit{h}\in {\mathfrak{C}}_0.$$

(32)

By Lemma 3, for $R\gg R_0$,

$$\underset{\partial B_R}{\int }\mathit{s}\left(\mathit{u}\right)=\underset{\partial B_R}{\int }\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_R=\mathit{0}.$$

Therefore, taking into account (27),

$$\underset{R^3}{\int }\mathit{f}=\underset{T_R}{\int }\mathit{f}=\underset{R}{\overset{2R}{\int }}{\vartheta }^{\prime }\left(r\right)dr\underset{\partial B_r}{\int }\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_r=\mathit{0},$$

(33)

Since

$$\mathcal{V}\left[f\right]\left(x\right)=\underset{R^3}{\int }\left[\mathcal{U}(x-y)-\mathcal{U}\left(y\right)\right]\mathit{f}\left(y\right)d{v}_y+\mathcal{U}\left(x\right)\underset{R^3}{\int }\mathit{f},$$

by (33), Lagrange’s theorem and (29)

$$\nabla \mathcal{V}\left[\mathit{f}\right]\left(x\right)=O\left(r^{-3}\right),$$

so that

$$\nabla \mathcal{V}\left[\mathit{f}\right]\in L^q,q\in (1,2].$$

(34)

Then, by (31), the map (30) is contractive for q in a right neighborhood of 1 so that

$$\mathit{u}\in D^{1,q}(\Omega ),q\in (1,2].$$

(35)

Conversely, if (35) holds, then a simple computation yields

$$\underset{\partial \Omega }{\int }\mathit{s}\left(\mathit{u}\right)=\underset{R^3}{\int }\mathbf{C}[\nabla \mathit{u}]\nabla g=-\frac{1}{R}\underset{T_R}{\int }\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_r,$$

(36)

where g is the function (24). By Hölder’s inequality,

$$\frac{1}{R}\left|\underset{T_R}{\int }\mathbf{C}[\nabla \mathit{u}]{\mathit{e}}_r\right|\le \frac{c}{R}{\left\{\underset{T_R}{\int }{|\nabla \mathit{u}|}^{3/2}\right\}}^{2/3}{\left\{\underset{T_R}{\int }dv\right\}}^{1/3}\le {\left\{\underset{T_R}{\int }{|\nabla \mathit{u}|}^{3/2}\right\}}^{2/3}.$$

Therefore, letting $R\to +\infty$ in (36) yields

$$\underset{\partial \Omega }{\int }\mathit{s}\left(\mathit{u}\right)=\mathit{0}$$

and this implies (32).

From

$$\begin{array}{cc}\hfill {\mathcal{V}}_i\left[f\right]\left(x\right)=& {\displaystyle \underset{R^3}{\int }\left[{\mathcal{U}}_{ij}(x-y)-\mathcal{U}\left(y\right)-y_k{\partial }_k{\mathcal{U}}_{ij}\left(y\right)\right]f_j\left(y\right)d{v}_y}\hfill \\ \hfill +& {\displaystyle {\mathcal{U}}_{ij}\left(x\right)\underset{R^3}{\int }{f}_j+{\partial }_k\mathcal{U}\left(x\right)\underset{R^3}{\int }{\mathcal{U}}_{ij}\left(y\right)f_j\left(y\right)d{v}_y}\hfill \end{array}$$

by (33), Lemma 4, Lagrange’s theorem and (29)

$$\nabla \mathcal{V}\left[\mathit{f}\right]\left(x\right)=O\left(r^{-4}\right),$$

so that $\nabla \mathcal{V}\left[\mathit{f}\right]\in L^1$. Since $\mathit{f}\in L^2\left(R^3\right)$ has compact support and satisfies (33), it belongs to ${\mathcal{H}}^1$ (see [7] p. 92) and by Lemma 2$\mathcal{V}\left[\mathit{f}\right]\in {\mathcal{H}}^1$. Hence, it follows that (30) maps ${\mathcal{H}}^1$ into itself and

$$\parallel {\mathit{v}}^{\prime }-{\mathit{z}}^{\prime }{\parallel }_{{\mathcal{H}}^1}\le \parallel \mathbf{C}-{\mathbf{C}}_0{\parallel }_{L^{\infty }\left(\complement S_{R_0}\right)}{\parallel \mathit{v}-\mathit{z}\parallel }_{{\mathcal{H}}^1}.$$

Since, by assumptions, $\parallel \mathbf{C}-{\mathbf{C}}_0{\parallel }_{L^{\infty }\left(\complement S_{R_0}\right)}$ is small, (30) is a contraction and by the above argument its (unique) fixed point must coincide with $\mathit{v}$ so that $\nabla \mathit{u}\in L^1(\Omega )$. □

We aim at concluding the paper with the following remarks.

(i)

It is evident that the hypothesis that $\mathbf{C}$ is regular at infinity can be replaced by the weaker one that $|\mathbf{C}-{\mathbf{C}}_0|$ is suitably small at a large spatial distance.

(ii)

The operator $\mathcal{V}$ maps boundedly the Hardy space ${\mathcal{H}}^q$$(q\in (0,1\left]\right)$ into itself [7]. Hence, the argument in the proof of (16) can be used to show that $\nabla \mathit{v}\in {\mathcal{H}}^q$, $q>3/4$. We can then use the Sobolev–Poincaré (see [8] p. 255) to see that $u\in L^q(\Omega )$ for $q>1$.

(iii)

Relation (16) is a kind of Stokes’ paradox in nonhomogeneous elastostatics: if $\mathbf{C}$ is regular at infinity, then the system

$$\begin{array}{cc}\hfill \mathrm{div}\mathbf{C}[\nabla \mathit{h}]& =\mathit{0}\mathrm{in}\Omega ,\hfill \\ \hfill \mathit{h}& =\tau \mathrm{on}\partial \Omega ,\hfill \\ \hfill {\displaystyle \underset{\partial B}{\int }\mathit{h}(R,\sigma )d\sigma }& =o\left(R^{-1}\right),\hfill \end{array}$$

with τ nonzero constant vector, does not admit solutions.

(iv)

If $\mathbf{C}$ is constant and strongly elliptic, then $\mathit{u}$ is analytic in $\Omega$ and at large spatial distance admits the representation

$$\mathit{u}\left(x\right)=\mathcal{U}\left(x\right)\underset{\partial \Omega }{\int }\mathit{s}\left(\mathit{u}\right)+\nabla \mathcal{U}\left(x\right)\underset{\partial \Omega }{\int }\left(\xi \otimes \mathit{s}\left(\mathit{u}\right)-\mathbf{C}[\widehat{\mathit{u}}\otimes \mathit{n}]\right)\left(\xi \right)+\psi \left(x\right)$$

with ${\left|\mathit{x}\right|}^3\left|\psi \left(x\right)\right|\le c$. Therefore, in the homogeneous case, the conclusions of Theorem 1 hold pointwise:

$${\left|\mathit{x}\right|}^2\left|\mathit{u}\left(x\right)\right|\le c⟺\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right)=\mathit{0},\forall \mathit{h}\in {\mathfrak{C}}_0,$$

$$\underset{\partial \Omega }{\int }\mathbf{C}[\widehat{\mathit{u}}\otimes \mathit{n}]=\mathit{0},\underset{\partial \Omega }{\int }\widehat{\mathit{u}}·\mathit{s}\left(\mathit{h}\right)=\mathit{0},\forall \mathit{h}\in \mathfrak{C}\Rightarrow {\left|\mathit{x}\right|}^3\left|\mathit{u}\left(x\right)\right|\le c.$$