Boundary value problems for the Lamé-Navier system in fractal domains

1.   Introduction

The displacement vector $\vec{u}$ of the points of a three-dimensional isotropic elastic body in the absence of body forces is described by the Lamé-Navier system

Here, the quantities $\mu > 0$ and $\lambda > -\frac{2}{3}\mu$ are the basic constants characterizing the elastic properties of the body (constants usually referred to as Lamé parameters ^[1,2]). For more details we refer to ^[3,4,5,6,7].

It has recently been shown in ^[8] that the Lamé equation (1.1) admits the form

where

stands for the Dirac operator constructed with the basis of the real Clifford algebra ${\mathbb R}_{0, 3}$. The elements in the kernel of $\partial_{\underline{x}}$ are called monogenic functions ^[9,10], which represent the main object of the so-called Clifford analysis.

Applications of this function theory to elastic materials are remarkable and have already been developed in ^{[11,12,13,14,15,16,17,18]}. More on these interesting topics the reader can find in the books ^[19,20]. It should be pointed out, however, that the study of boundary value problems for such physical models has been confined to smoothly bounded domains, since there exist enough obstacles to a rigorous treatment of such problems in the more challenging case of domains with fractal boundaries. From the point of view of engineering applications considerable interest attaches to the solution of more general problems when the body under consideration admits a boundary of more general character.

Fractals are not only relevant from a mathematical point of view, but also have important applications and are widely used in physics, biology, pharmaceutical sciences and chemistry ^{[21,22,23,24,25]}. It is for these reasons that it is not unreasonable to consider the above problems under such a general geometric conditions.

In this paper we make essential use of the methods introduced in ^[26,27] to derive a representation formula for the solutions (1.1) and its applications to boundary value problems for such a system in a very wide classes of regions. We stress that our approach allows domains with fractal boundaries, a question that as far as we know has not been considered before. The present work represents a three-dimensional generalization of the recently published paper ^[28], where the Lamé system is considered on plane domains with fractal boundary, using classical complex analysis techniques.

2.   Preliminaries

Let $e_1, e_2, e_3$ be an orthonormal basis of ${\mathbb R}^3$, with the multiplication rules

In this way, the Euclidean space

is embedded in the real Clifford algebra ${\mathbb R}_{0, 3}$ generated by $e_1, e_2, e_3$ over the field of real numbers ${\mathbb R}$.

An element $a\in {\mathbb R}_{0, 3}$ may be written as $a = \sum_{A} a_A e_A$, where $a_A$ are real constants and $A$ runs over all the possible ordered sets

and

The scalar part of $a$ is defined by${Sc}[a]: = a_0$.

The product of two Clifford vectors admits the splitting

where

is a scalar, while

is a $2$-vector.

In general, we will consider functions defined on subsets of ${\mathbb R}^{3}$ and taking values in ${\mathbb R}_{0, 3}$, which can be written as $f = \sum_{A} f_A e_A$, the $f_A$'s being ${\mathbb R}$-valued functions.

The spaces of all $k$-time continuous differentiable and $p$-integrable functions are component-wise defined and denoted by $C^k({\bf E})$ and $L^p({\bf E})$ respectively, where ${\bf E}\subset {\mathbb R}^{3}$.

The Dirac operator $\partial_{\underline{x}}$ in ${\mathbb R}^{3}$ is defined for $C^1$-functions as

This operator allows a factorization of the Laplacian $\triangle$ in ${\mathbb R}^3$, namely

The fundamental solution of $\triangle$ is given by

where $\sigma_{3}$ denotes the surface area of the unit sphere in ${\mathbb R}^3$.

The so-called Clifford-Cauchy kernel is then constructed as

which satisfies the equations $\partial_{\underline{x}} E_0 = E_0 \partial_{\underline{x}} = 0$ in ${\mathbb R}^{3}\setminus\{0\}$.

The ${\mathbb R}_{0, 3}$-valued solutions of $\partial_{\underline{x}} f = 0$ ($f \partial_{\underline{x}} = 0$) are called left monogenic (right monogenic) functions. Those functions which simultaneously satisfy both equations are referred as two-sided monogenic.

Unless stated otherwise, we always suppose that $\Omega$ is a smoothly bounded Jordan domain of ${\mathbb R}^3$. Later, the above smoothness assumption will be completely relaxed including the general case of a fractal boundary. In the sequel, the following notation will be used for the interior and exterior domains: $\Omega_+: = \Omega$, $\Omega_-: = {\mathbb R}^3\setminus\overline{\Omega}$.

The Cliffordian-Stokes theorem ^[9] leads to the Borel-Pompeiu integral representation formula for ${\mathbb R}_{0, 3}$-valued functions $f\in C^1(\Omega\cup\Gamma)$. Namely,

where

and

are, respectively, the Cauchy and Teodorescu transforms of $\varphi$.

Hereby $\underline{n}(\underline{y})$ is the outward normal at $\underline{y}\in\Gamma$, and $dS$ ($dV$) denotes the surface (volume) measure.

In particular, for left monogenic functions one has in $\Omega$

Right-handed versions of formulas (2.1) and (2.2) are similarly obtained by using the integral transforms

and

The inframonogenic functions have been introduced in ^[29,30] (see also ^[8,26,27]) as the ${\mathbb R}_{0, 3}$-valued solutions of the sandwich equation

Such functions represent a refinement of the more traditional biharmonic functions, see for instance ^[31,32].

As proved in ^[26] any function $f$ in $C^1(\Omega)$, inframonogenic in $\Omega$, can be represented by

and the operator

runs as a right inverse of the sandwich operator $\partial_{\underline{x}}\cdot \partial_{\underline{x}}$, i.e. $\partial_{\underline{x}} {\cal T}_\Omega^{ \bf{infra}}f \partial_{\underline{x}} = f$.

3.   Boundary value problems for the Lamé-Navier system

We start this section by considering a Clifford reformulation of the Lamé system (1.1) obtained in ^[8].

By the use of the identities

and

the system (1.1) becomes

where $\alpha = \frac{\mu+\lambda}{2}$, $\beta = \frac{3\mu+\lambda}{2}$.

This implies immediately the factorization ${\cal L}_{\alpha, \beta}\vec{u} = \partial_{\underline{x}} \partial_{\underline{x}}^{\alpha, \, \beta}\vec{u}$, where

is a first-order Dirac type operator introduced and studied recently in ^[33].

As already mentioned in ^[33] an analogous factorization for the Lamé system is derived in [19,p. 85], where use has been made of the operator $M^{-1}f = \frac{\lambda+2\mu}{\mu}f_0+ \underline{f}$. Indeed, we have

This approach allows the entry of quaternionic analysis techniques in obtaining integral representation formula for the solution of (1.1) as the composition of Teodorescu and Cauchy transforms.

The idea of the present paper is more in the direction of ^[26,33], where explicit integral representation formulas are obtained in terms of properly defined Cauchy and Teodorescu transforms, this time closely related to the Lamé-Navier operator ${\cal L}_{\alpha, \beta}$. Our method can be extended without difficulty to the multidimensional elasticity theory ^[1,34].

It is easily seen that $\alpha\not = 0$ and $\beta\not = 0$. This follows from the conditions $\mu > 0$ and $\lambda > -\frac{2}{3}\mu$.

The Dirichlet problem for the system of elastostatics in a Lipschitz bounded domain $\Omega\subset {\mathbb R}^3$ with boundary $\Gamma$:

was considered, for example, in ^[2].

It will be seen now how the above Clifford reformulation offers the possibility of proving in a very simple manner the following uniqueness theorem. Compare with [19,Theorem 4.3.3].

Theorem 1. Let be $\vec{f}\in C(\Gamma)$. If a solution of the Dirichlet problem (3.2) exists in $C^2(\Omega)\cap C^1(\overline{\Omega})$, the solution is unique.

Proof.

As usually, we are reduced to prove that the problem

allows only the null solution $\vec{u}\equiv 0$.

By means of the Stokes formula we have

which gives

and finally

Since $\underline{u}(\underline{y}) \partial_{\underline{y}} = -{\rm{div}} \underline{u}(\underline{y})- {{{\rm{rot}}}} \underline{u}(\underline{y})$, it follows that

and moreover

Taking the scalar part in (3.4) yields

If $\beta\leq\alpha$, then $3\mu+\lambda\leq\mu+\lambda$ and $2\mu\leq 0$, the later being false, since $\mu > 0$. Similarly, the assumption $\beta\leq -\alpha$ leads to a contradiction with $\frac{\lambda}{\mu} > -\frac{2}{3}$.

Therefore

and hence

which together with the boundary condition $\underline{u} = 0$, completes the proof.

We are rather interested in the investigation of the jump problem for the Lamé system (1.1):

where $\vec{u}^\pm(\underline{x})$ are the limit values of $\vec{u}$ at the point $\underline{x}\in\Gamma$ as this point is approached from $\Omega_{\pm}$, respectively.

The vector valued function $\vec{f}$ is assumed to be in the higher order Lipschitz class ${{\rm{Lip}}}(1+\alpha, \Gamma)$, $0 < \alpha < 1$; i.e. for each real component $f_i$, $i = 1, 2, 3$, of $\vec{f}$ there exists a collection $\{f_i^{(j)}, \, 0\le|j|\le 1\}$ of real uniformly bounded functions on $\Gamma$, with $f_i^{(0)} = f_i$, and so that

satisfies

Following [35,Theorem 4,page 177], any function $\vec{f}$ in ${{\rm{Lip}}}(1+\alpha, \Gamma)$ can be extended to the whole ${\mathbb R}^3$ as a continuously differentiable function, with the abuse of notation again denoted by $\vec{f}$. The Whitney extension $\vec{f}$ has $\alpha$-Hölder continuous partial derivatives, and moreover

for $|(j)| = 2$ and $\underline{x}\in {\mathbb R}^3\setminus\Gamma$. It will be assumed in the sequel that $c$ is a positive constant, which may have different values at different occurrences.

The above problem will be studied both for the more standard case of sufficiently smooth surfaces as well as for the pathological situation of considering domains with fractal boundary.

4.   Smooth boundary case

Before further development of problem (3.6) let us state and prove a sort of Borel-Pompeiu representation formula in terms of the Lamé operator ${\cal L}_{\alpha, \beta}$. Introduce the notation

where

Let us first prove that ${\cal T}_\Omega^{ {\cal L}}$ works as an inverse operator for ${\cal L}_{\alpha, \beta}$. For that there is no restriction on $\Omega$ other than the requirement of being open and bounded.

Theorem 2. Let be $\vec{f}\in C^2(\Omega),$ then

Proof.

We restrict our consideration to $\underline{x}\in\Omega_+$. The case $\underline{x}\in\Omega_-$ does not meet with any essentially new difficulties.

Taking into account that

we have

where we used the identities $\beta^*\beta+\alpha^*\alpha = 1$ and $\alpha\beta^*+\beta\alpha^* = 0$.

Theorem 3. Let $\vec{f}\in C^{2}(\Omega)\cap C^1(\overline{\Omega})$. Then, for $\underline{x}\in\Omega$ we have

We need first the following auxiliary result.

Lemma 4.

Proof.

The proof is quite similar to that of [26,Theorem 3.1] and requires the use of the identity

Proof of Theorem 3

It follows that

Now we make use of both, Lemma 4, [26,Theorem 3.1] and the iterated Borel-Pompeiu formula associated to $\partial_{\underline{x}}^2\vec{f}$ (see ^[36,37])

Consequently,

Since

and

after subtracting them we obtain

and finally

Corollary 5. Let $\vec{f}\in C^{2}(\Omega)\cap C^1(\overline{\Omega})$. If, moreover, $\vec{f}$ satisfies (3.1) then in $\Omega$ we have

Let now $\vec{f}$ be intrinsically defined as a $C^1$-smooth function on $\Gamma$. A direct but non-trivial calculation shows that a function given by

satisfies the Lamé system in ${\mathbb R}^3\setminus\Gamma$.

Now we are able to characterize the solvability of Problem (3.6).

Theorem 6. Let $\vec{f}\in {\mathit{{\rm{Lip}}}}(1+\alpha, \Gamma)$. Then a solution of (3.6) is given by

Moreover, it is unique under the vanishing conditions $\vec{u}(\infty) = \vec{u} \partial_{\underline{x}} (\infty) = 0$.

Proof.

Using the formulae of Plemelj–Sokhotski ^[9] we conclude that the first two summands in (4.1) have by passage through $\Gamma$ the jump $\beta^*\beta f(\underline{x})$ and $\alpha^*\alpha f(\underline{x})$, respectively. Since the remaining terms are weekly–singular parametric integrals, they have no jumps through $\Gamma$. Consequently,

The proof of the second jump condition needs some more calculations. Indeed, we have in ${\mathbb R}^3\setminus\Gamma$ that

After using $\alpha^*\beta = -\beta^*\alpha$, we obtain

As usual let $\chi_\Omega$ be the characteristic function of $\Omega$. Using again the iterated Borel-Pompeiu formula associated to $\partial_{\underline{x}}^2\vec{f}$ (see ^[36,37]),

we get

which, after applying in the right-hand side the Borel-Pompeiu formula associated to $\partial_{\underline{x}}$, gives

Finally we have

from which the second boundary condition in (3.6) is again a direct consequence of the Plemelj-Sokhotski formulae.

The proof of the uniqueness can be done indirectly. Assume that $\vec{u}_1, \vec{u}_2$ are two solutions of (3.6), then it implies that $\vec{\omega} = \vec{u}_1-\vec{u}_2$ fulfills

Let us prove that $\vec{\omega}\equiv 0$.

Since $\vec{\omega}$ satisfies ${\cal L}_{\alpha, \beta}\vec{\omega}(\underline{x}) = 0$ in ${\mathbb R}^3\setminus\Gamma$, the function ${\varpi} = \vec{\omega} \partial_{\underline{x}}$ satisfies the equation $\alpha \partial_{\underline{x}} {\varpi}+\beta {\varpi} \partial_{\underline{x}} = 0$ there. From the previous statement it is easy to show that the ${\mathbb R}_{0, 3}$-valued function ${\varpi}_* = {\varpi}_0+\frac{\beta-\alpha}{\beta+\alpha}\underline{ {\varpi}}$ is (left) monogenic in ${\mathbb R}^3\setminus\Gamma$.

Hence, the auxiliary function ${\varpi}_*$ is a solution of the boundary value problem

From the Painlevé and Liouville theorems in Clifford analysis ^[9] it follows that the above problem has the unique trivial solution $\phi\equiv 0$, so we have ${\varpi}_*\equiv 0$ and hence ${\varpi}\equiv 0$ in ${\mathbb R}^3$.

Consequently, $\vec{\omega}$ is (right) monogenic in ${\mathbb R}^3\setminus\Gamma$, vanishes at $\infty$ and has no jump through $\Gamma$. Finally, a repeated use of the Painlevé and Liouville theorems yields $\vec{\omega}\equiv 0$ and we are done.

5.   Fractal boundary case

The main new ingredient of this section is the extension of our previous considerations to the case of domains $\Omega$ admitting a fractal boundary. We follow ^[38] in assuming that $\Gamma$ is $d$-summable for $2 < d < 3$, which means that the integral

exists in the improper sense. Here$N_\Gamma(\tau)$ denotes the least number of balls of radius $\tau$ needed to cover $\Gamma$.

As was early remarked in ^[38] any surface $\Gamma$ with fractal box dimension ${\rm D}(\Gamma)$ is $d$-summable for any $d = {\rm D}(\Gamma)+\epsilon$, $\epsilon > 0$.

The following result was proved in ^[38] and it is really the heart of the proof of the main theorem of this section.

Lemma 7. ^[38] If $\Omega$ is a Jordan domain of ${\mathbb R}^{3}$ and its boundary $\Gamma$ is $d$-summable, then the expression $\sum_{Q\in{\mathcal{W}}}|Q|^d$, called the $d$-sum of the Whitney decomposition ${\mathcal W}$ of $\Omega$, is finite.

Recall that the Whitney decomposition of $\Omega$ involves a collection of disjoint cubes, whose lengths are proportional to their distance from $\Gamma$. For details we refer the reader to ^[35].

To solve the problem (3.6) in the fractal setting, we need first a few results.

Lemma 8. Let $\vec{f}\in {\mathit{{\rm{Lip}}}}(1+\alpha, \Gamma)$, then ${\cal L}_{\alpha, \beta}[\vec{f}]\in L^p(\Omega)$ for any $p\le\frac{3-d}{1-\alpha}$.

Proof.

From (3.9), we have $| {\cal L}_{\alpha, \beta}\vec{f}(\underline{x})| \leqslant c \, dist(\underline{x}, \Gamma)^{\alpha-1}$ for $\underline{x}\in\Omega$. After such estimate, the statement can be proved quite analogously to [39,Lemma 4.1].

Lemma 9. Let $\vec{f}\in {\mathit{{\rm{Lip}}}}(1+\alpha, \Gamma)$ with $\alpha > \frac{d}{3}$. Then the functions ${\cal T}_\Omega^{ {\cal L}}\big({\cal L}_{\alpha, \beta}\vec{f}\big)$ and $\big[ {\cal T}_\Omega^{ {\cal L}}({\cal L}_{\alpha, \beta}\vec{f})\big] \partial_{\underline{x}}$ are continuous in ${\mathbb R}^3$.

Proof.

Let $\vec{\varphi}: = {\cal L}_{\alpha, \beta}[\vec{f}]$ and prove first the continuity of ${\cal T}_\Omega^{ {\cal L}}\vec{\varphi}$. Indeed, take $\underline{x}, \underline{z}\in {\mathbb R}^3$, then

If follows that

Then

It follows from $\alpha > \frac{d}{3}$ and Lemma 8 that $\vec{\varphi}$ is integrable in $\Omega$. Consequently, every integral in (5.1) is finite and hence

goes to $0$ as $\underline{x}\to \underline{z}$.

On the other hand,

Because of

it follows that

At this point we use again the integrability of $\vec{\varphi}$ together with the above inequality to see that

goes to $0$ as $\underline{x}\to \underline{z}$, which summarizing proves the continuity of ${\cal T}_\Omega^{ {\cal L}}\vec{\varphi}$.

To prove the continuity of $\big[ {\cal T}_\Omega^{ {\cal L}}({\cal L}_{\alpha, \beta}\vec{f})\big] \partial_{\underline{x}}$ we use the identity

and the following one proved in [26,Theorem 4.1]:

Therefore, we have

According to the condition $\alpha > \frac{d}{3}$ and Lemma 8, we conclude that $\vec{\varphi}\in L^p(\Omega)$ with $p = \frac{3-d}{1-\alpha} > 3$. Now the assertion is proved by appealing to [10,Proposition 8.1].

Let us come back to the task of finding a solution of (3.6) in our general geometric context.

Theorem 10. Let $\vec{f}\in {\mathit{{\rm{Lip}}}}(1+\alpha, \Gamma)$. Under the assumption $\alpha > \frac{d}{3}$ the problem (3.6) has a solution given by

Proof.

It is easy to verify that $\vec{u}$ satisfies ${\cal L}_{\alpha, \beta}\vec{u} = 0$ in ${\mathbb R}^3\setminus\Gamma$, which follows from Theorem 2. Moreover, the validity of the boundary conditions in (3.6) is straightforwardly implied by Lemma 9.

Remark 11. As we have seen already with the case of sufficiently smooth boundaries, the uniqueness of the solution of (3.6) is directly related with the removability of $\Gamma$ for continuous monogenic functions. Although this result is no longer available in general, nevertheless a Dolzhenko theorem proved in ^[40] is instead more appropriate to deal with the picture of uniqueness in the case of a $d$-summable boundary $\Gamma$, see [39,Theorem 4.2]. Due to the deep similarity we will omit the details.

6.   Conclusions

Consider a vector field $\vec{u}\in C^2(\Omega)\cap C^1(\overline{\Omega})$, which is a solution of the Lamé-Navier system ${\cal L}_{\alpha, \beta}\vec{u} = 0$ in $\Omega$. We have shown that $\vec{u}$ admits in $\Omega$ an integral representation formula in terms of its boundary values and those of their first order partial derivatives. We also provide a particular solution of the inhomogeneous Lamé-Navier system ${\cal L}_{\alpha, \beta}\vec{u} = f$ by means of the generalized Teodorescu transform ${\cal T}_\Omega^{ {\cal L}}\vec{f}$. The above results are applied to obtain an explicit solution of boundary value problems for such a system in a very wide class of bounded domains in ${\mathbb R}^3$, .

Acknowledgments

This paper was supported in part by a grant from the Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain.