The existence of an energy minimizing configuration for multiple solid objects floating in a bath of three liquids

Abstract

We consider the physical configuration of a container which holds a finite number of movable solid objects and three immiscible fluids. Each fluid volume is prescribed, and the container is completely filled. The configuration is modeled using sets of finite perimeter, and the energy of the configuration is given using the theory of functions of bounded variation. We show that there exists an admissible configuration that attains the minimum energy.

1 Introduction

We consider a bounded container $\Omega$ to be a simply connected domain that is bounded in ${\mathbb {R}}^n$. We will present results for a Lipschitz domain $\Omega$; however, some of our hypotheses will benefit from showing that they are physically attainable from given criterion. For that reason, we will sometimes discuss the case when the container is bounded and cylindrical: $\Omega := G \times (0, p)$ where G is a bounded, simply connected domain in ${\mathbb {R}}^{n-1}$.

Into our container, we place a finite number of rigid but movable bodies ${\mathcal {B}}_k$, with $k=1,2,\ldots K$. We take each ${\mathcal {B}}_k$ to be the closure of an open set, with ${\mathcal {B}}^\circ _k\subset \Omega$ for $k=1,2,\ldots K$. We refer to the collection of ${\mathcal {B}}_k$ as . These bodies have densities $\rho _{{\mathcal {B}}_k}$.

The remainder of $\Omega$ is filled with three fluids, denoted by the sets $E_i$ where $i = 1, 2, 3$. See Fig. 1 for an example. We assume some structure on these sets, in that we require each $E_i$ to be a Caccioppoli set. We recall that if $E\subset \Omega$ is a Caccioppoli set, otherwise known as a set of finite perimeter, then the characteristic function of E, denoted by $\phi _E$, is in ${\mathrm{BV}}(\Omega )$, which is the class of functions of bounded variation on $\Omega$. This allows us to measure the perimeter in terms of the total variation of the characteristic function of each set. To be precise, given a domain U, and a Caccioppoli set E, we have

$$\begin{aligned} P(E,U) := \int _U\left| D\phi _E\right| := {\mathrm{sup}}\left\{ \int _U\phi _E\,\text {div }g\, {\mathrm{d}}x : g \in C_c^1 (U; {\mathbb {R}}^n), ||g||_{C^0} \le 1\right\} . \end{aligned}$$

Fig. 1

A container with three fluids and one floating object

Naturally, the density of each fluid is prescribed and given by the quantity $\rho _{E_i}$. We also use the notation . The interface between fluids has an associated surface tension, and we denote these quantities by $\sigma _{ij}$, which are symmetric in i and j, corresponding to the interface between fluids $E_i$ and $E_j$. We define

$$\begin{aligned} \alpha _i := \frac{1}{2}(\sigma _{ij} + \sigma _{ik} - \sigma _{jk}), \end{aligned}$$

(1)

where $i, j, k = 1, 2, 3$ taken to be mutually distinct, so that we can use the perimeters of the sets $E_i$ to describe the surface tensions on the interfaces $S_{ij} := \partial ^*E_i\cap \partial ^*E_j$. Here, we use $\partial ^*E$ to denote the reduced boundary of a set of finite perimeter. (For background on the reduced boundary of a set and functions of bounded variation in general, see Massari and Miranda [6]. We select this particular treatment of BV theory from the growing list of good books on the topic as it uniquely contains some particular material specifically related to our project.) We observe that $\alpha _i + \alpha _j = \sigma _{ij}$, and for a given domain U, we have

$$\begin{aligned} \sum \alpha _i \int _U |D\phi _{E_i}| = \sum \sigma _{ij}|S_{ij}\cap U|. \end{aligned}$$

We will also require that each body ${\mathcal {B}}_k$ is represented by a Caccioppoli set. Each ${\mathcal {B}}_k$ may be subjected to rigid motions and can be described as ${\mathcal {B}}_k = {\mathcal {B}}_k(c_k, R_k) := \{ y = c_k + R_kx : x \in {\mathcal {B}}_{k,0}\}$, where an initial description of each shape is given by ${\mathcal {B}}_{k,0}$. Here, $c_k \in {\mathbb {R}}^n$ denotes a translation and $R_k = R_k(d, \theta ) \in SO(n)$ describes a rotation with respect to some axis generated by a unit vector d and about some angle $\theta \in [0,\pi ]$. The quantities $c_k$ and $R_k$ are restricted by two assumptions. First, we assume that ${\mathcal {B}}^\circ _k\subset \Omega$, though we allow the possibility that $\partial \Omega \cap {\mathcal {B}}_k \ne \emptyset$. Further, we assume that there are a finite number of contact points: and are both finite collections of points. At these contact points between the bodies, we allow ${\mathcal {B}}_k\cap {\mathcal {B}}_\ell = \{p_m\}$ for a finite number of points $p_m\in {\mathbb {R}}^n$ when $k\ne \ell$. We do, however, make the assumption that the interiors of the bodies do not intersect. It follows that each is measurable, and we denote $V_i = |E_i|$ to be the volume of that fluid. We will use the notation $(x', x_n)$ for $x \in {\mathbb {R}}^n$.

The energy of our configuration can then be described by the functional

(2)

The integrals over $\partial \Omega$ and measure the wetting energy along the container and the solids, respectively, and the remaining integrals measure the gravitational potential energy. Here, we use the gravitational constant g, and the adhesion energy between the fluid $E_i$ and $\partial \Omega$ is given by $\beta _i$, while the adhesion energy between the fluid $E_i$ and each $\partial {\mathcal {B}}_k$ is given by $\tau _{ik}$. We seek a configuration that minimizes ${\mathcal {F}}$ over the class

(3)

The results and approach in this paper are generalizations of two previous works. The first is a proof given by Massari [4] of the existence of a minimizer treating the case where there are no floating objects, and the second is an approach by Bemelmans, Galdi, and Kyed that spans (parts of) two papers, [1, 2], treating the case where there are two fluids and one floating object. These results are in turn generally based on the proof of existence of a minimizer for the two-fluid, no-solid problem given by Emmer [3], which can also be read about in [6].

2 Main result

We begin by establishing a lower bound on ${\mathcal {F}}$ and then forming a minimizing sequence of . We then prove that ${\mathcal {F}}$ is lower semicontinuous with respect to this sequence. Finally, we will prove the existence of an energy minimizer of ${\mathcal {F}}$ in the class ${\mathcal {C}}$.

We have required that the bodies ${\mathcal {B}}_k$ and ${\mathcal {B}}_\ell$ contact at most at a finite number of points. As in [2], we can use conditions to enforce a stricter geometric requirement if we consider the bounded cylindrical container $\Omega = G\times (0,p)$ with $\partial G$ and each $\partial {\mathcal {B}}_k$ to be of class $C^2$ and ${\mathcal {B}}_k(c_k, R_k)$ to have a projection $P_k(c_k, R_k)$ into G such that

$$\begin{aligned} \min _{R \in SO(3), x' \in \partial P_k(c_k, R_k)} K(\partial P_k(c_k, R_k), x') > \max _{x' \in \partial \Omega } K(\partial \Omega , x'). \end{aligned}$$

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This restriction ensures that all possible translations and rotations of each ${\mathcal {B}}_k$ give projections $P_k(c_k,R_k)$ with curvature strictly larger than that of G, so if a solid and container touch, then they only touch at a single point $p_k$. This also implies that the bodies ${\mathcal {B}}_k$ and ${\mathcal {B}}_\ell$ contact at most one point. That there are a finite number of contact points is a crucial condition, and we proceed with this assumed to be the case, no matter what restrictions led to this condition.

Fig. 2

Detail near a contact point

We next construct open (almost) neighborhoods of $\partial \Omega$ and $\partial {\mathcal {B}}_k$ in strips that “dimple” near each contact point. See Fig. 2. We first consider contact points $p_m\in \partial \Omega$, where the index m ranges over all of the contact points with the container boundary. Then, for a neighborhood $U_{\epsilon _m}$ of $p_m$, we can define a tangent plane $\Pi$ and a neighborhood $A_{\epsilon _m}\subset \Pi$ at $p_m$. We can then describe the boundaries $\partial \Omega$ and $\partial {\mathcal {B}}_k$ by the graphs of the respective functions $y_n = \omega _m(y^\prime )$ and $y_n = \beta _m(y^\prime )$, where $y^\prime \in A_{\epsilon _m}\subset \Pi$ are coordinates in the tangent plane at $p_m$, centered at $y_0' = 0$. We restrict our neighborhood so that $A_{\epsilon _m}$ is the domain of both $\omega _m$ and $\beta _m$. We then have $\beta _m(y') > \omega _m(y')$ for all nonzero $y' \in A_{\epsilon _m}$. (We are locally using $\beta _m$ and $\tau _k$ below as functions, not to be confused with the wetting energies $\beta _i$ and $\tau _{ik}$.)

Next, we consider contact points $p_\ell \notin \partial \Omega$, where the index $\ell$ ranges over all contact points between bodies. We will use the convention that the contact point is on ${\mathcal {B}}_\ell$ and occurs with some other bodies ${\mathcal {B}}_k$. Then, for a neighborhood $U_{\epsilon ^\ell _k}$ of $p_\ell$, we can define a tangent plane $\Pi$ and a neighborhood $A_{\epsilon ^\ell _k}\subset \Pi$ at $p_\ell$. We can then describe the boundaries $\partial {\mathcal {B}}_\ell$ and $\partial {\mathcal {B}}_k$ by the graphs of the respective functions $y_n = \beta _\ell (y^\prime )$ and $y_n = \beta _k(y^\prime )$, where $y^\prime \in A_{\epsilon ^\ell _k}\subset \Pi$ are coordinates in the tangent plane to $p_\ell$, centered at $y_0' = 0$. As before, we restrict our neighborhood so that $A_{\epsilon ^\ell _k}$ is the domain of both $\beta _\ell$ and $\beta _k$. We then have $\beta _k(y') > \beta _\ell (y')$ for all nonzero $y' \in A_{\epsilon ^\ell _k}$.

If $\Omega$ or are not smooth enough to uniquely define tangent planes, in place of $\Pi$, we choose one of the supporting planes for $\Omega$ or ${\mathcal {B}}_\ell$ that is also a supporting plane for ${\mathcal {B}}_k$. Since all of these objects are Lipschitz, we can choose such a plane so that the above discussion still holds.

Fix $\epsilon > 0$. First, define the following for each $p_m$

$$\begin{aligned} \tau _m(p'):= & {} {\left\{ \begin{array}{ll} \frac{\beta _m(p') - \omega _m(p')}{3\epsilon }, &{} \text { if } \beta _m(p') - \omega _m(p') < 3\epsilon , \\ 1, &{} \text { if } \beta _m(p') - \omega _m(p') \ge 3\epsilon , \end{array}\right. } \end{aligned}$$

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$$\begin{aligned} \beta _m^*(\delta ):= & {} \{p = (p', p_n) : \,\, p_n = \beta _m(p') - \delta \tau _m(y)\}, \end{aligned}$$

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$$\begin{aligned} \omega _m^*(\delta ):= & {} \{p = (p', p_n) : \,\, p_n = \omega _m(p') - \delta \tau _m(y)\}, \end{aligned}$$

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$$\begin{aligned} ({\mathcal {B}}^0_m)_\epsilon ^*:= & {} \cup _{\delta \in (0, \epsilon )} \beta _m^*(\delta ), \end{aligned}$$

(8)

$$\begin{aligned} (\Omega _m)_\epsilon ^*:= & {} \cup _{\delta \in (0, \epsilon )} \omega _m^*(\delta ). \end{aligned}$$

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Then, define the following for each $p_\ell$

$$\begin{aligned} \tau _k^\ell (p'):= & {} {\left\{ \begin{array}{ll} \frac{\beta _k(p') - \beta _\ell (p')}{3\epsilon }, &{} \text { if } \beta _k(p') - \beta _\ell (p') < 3\epsilon , \\ 1, &{} \text { if } \beta _k(p') - \beta _\ell (p') \ge 3\epsilon , \end{array}\right. } \end{aligned}$$

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$$\begin{aligned} \beta _{k,\ell }^*(\delta ):= & {} \{p = (p', p_n) : \,\, p_n = \beta _k(p') - \delta \tau ^\ell _k(y)\}, \end{aligned}$$

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$$\begin{aligned} \beta _{\ell ,k}^*(\delta ):= & {} \{p = (p', p_n) : \,\, p_n = \beta _\ell (p') - \delta \tau ^\ell _k(y)\}, \end{aligned}$$

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$$\begin{aligned} ({\mathcal {B}}_{k,\ell })_\epsilon ^*:= & {} \cup _{\delta \in (0, \epsilon )} \beta _{k,\ell }^*(\delta ), \end{aligned}$$

(13)

$$\begin{aligned} ({\mathcal {B}}_{\ell ,k})_\epsilon ^*:= & {} \cup _{\delta \in (0, \epsilon )} \beta _{\ell ,k}^*(\delta ). \end{aligned}$$

(14)

Finally, we take intersections of these sets to obtain dimpled open neighborhoods of the boundaries of the container and each body. We define

$$\begin{aligned} ({\mathcal {B}}_k)_\epsilon ^*:= & {} \left( \cap (({\mathcal {B}}_k)^0_m)_\epsilon ^*\right) \cap \left( \cap ({\mathcal {B}}_{k,\ell })_\epsilon ^* \right) , \end{aligned}$$

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$$\begin{aligned} \Omega _\epsilon ^*:= & {} \cap (\Omega _m)_\epsilon ^*, \end{aligned}$$

(17)

where we have made use of a convention that if a contact point is not nearby, the neighborhood has width $\epsilon$ in the sense that given a domain U, $U_\epsilon := \{x\in U : {\mathrm{dist}}(x,\partial U) < \epsilon \}$.

Bemelmans et al. [2] proved a version of the following lemma, and their proof holds on our additionally restricted sets.

Lemma 2.1

Let $\partial \Omega$ and touch in at most a finite number of points. Then, for , we have

$$\begin{aligned}&\int _{\partial \Omega } u\, {\mathrm{d}}{\mathcal {H}}^{n-1} \le \sqrt{1+L^2} \int _{\Omega _\epsilon ^*} \left| Du\right| + c \int _{\Omega _\epsilon ^*} u\, {\mathrm{d}}x, \end{aligned}$$

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$$\begin{aligned}&\int _{\partial {\mathcal {B}}_k} u\, {\mathrm{d}}{\mathcal {H}}^{n-1} \le \sqrt{1+L^2} \int _{({\mathcal {B}}_k)_\epsilon ^*} \left| Du\right| + c \int _{({\mathcal {B}}_k)_\epsilon ^*} u\, {\mathrm{d}}x, \end{aligned}$$

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where c is a constant, L is an upper bound for $\vert D\omega _m\vert$ and $\vert D\beta _m\vert$ in $A_{\epsilon _m}$ or $\vert D\beta _\ell \vert$ and $\vert D\beta _k\vert$ in $A_{\epsilon ^\ell _k}$ when the objects are smooth, and L is the Lipschitz constant otherwise.

They also make the following applicable remark as to the usage of their lemma:

Under the assumption that $\partial {\mathcal {B}}_k$ and $\partial \Omega$ are regular surfaces, L can be made small by covering $\partial {\mathcal {B}}_k$ and $\partial \Omega$ by small open sets near each $p_k$ or $p_{\ell }$. Hence, the value of L is no extra restriction on the other physical parameters under that consideration.

We should point out that these types of inequalities also hold (for $L=0$) when the container has an interior sphere condition, which is certainly satisfied when the boundary of the domain is regular. See Tamanini [5].

Next, we establish a lower bound on ${\mathcal {F}}$.

Lemma 2.2

Let $\alpha _i, \rho _i, \rho _{{\mathcal {B}}_k} \in {\mathbb {R}}$ be such that $\rho _i \ge 0$, $\rho _{{\mathcal {B}}_k} \ge 0$, $\alpha _i \ge 0$, $\alpha _i + \alpha _j > 0$. Then, ${\mathcal {F}}$ is bounded below.

Proof

Since $\Omega$ is a bounded set in ${\mathbb {R}}^n$, we can define $m := \min \{x_n: (x^\prime ,x_n)\in \Omega \}$. Set $d := \max \{\rho _{E_i},\rho _{{\mathcal {B}}_k}\}$ Then, we can bound all of the terms related to gravity below by the quantity $gdm|\Omega |$. Similarly, we can completely wet the boundary $\partial \Omega$ with the set $E_i$ that corresponds to the smallest $\beta _i$, and we can completely wet each body ${\mathcal {B}}_k$ with the set $E_i$ corresponding to the smallest $\tau _{ik}$. Then, it only remains to bound the terms corresponding to interfaces between fluids. It is clear that these terms are bounded below by 0 if and only if each pair $\alpha _i + \alpha _j > 0$. Therefore, ${\mathcal {F}}$ is bounded below. $\square$

We will break the energy functional into parts. The gravity terms are more well behaved than the free surface and wetting terms, so we postpone their consideration and define

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The next lemma is the key technical result of this paper.

Lemma 2.3

Let $\alpha _i, \rho _i, \rho _{{\mathcal {B}}_k} \in {\mathbb {R}}$ be such that $\rho _i \ge 0$, $\rho _{{\mathcal {B}}_k} \ge 0$, $\alpha _i \ge 0$, $\alpha _i + \alpha _j > 0$. We also require two conditions which include a Lipschitz constant L:

$$\begin{aligned} \alpha _i + \alpha _j \ge \sqrt{1+L^2}|\beta _i - \beta _j|, \end{aligned}$$

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and

$$\begin{aligned} \alpha _i + \alpha _j \ge \sqrt{1+L^2}|\tau _{ik} - \tau _{jk}|, \end{aligned}$$

(22)

for each k. In addition suppose that we have a sequence and a configuration with in the sense of characteristic functions converging in $L^1(\Omega )$. Then, .

Proof

In order to show our desired result, we prefer to work with the equivalent form of the conclusion given by

We are free to choose one ordering of the sets, and we fix the convention that the $\beta _i$ terms are labeled in increasing order, i.e., $\beta _1 \le \beta _2 \le \beta _3$. We will also need an ordering on each of the $\tau _{ik}$ terms, so we introduce permutation functions $p_k(i) = j$ so that $\tau _{p_k(1)k}\le \tau _{p_k(2)k}\le \tau _{p_k(3)k}$ for each k.

The first technicality is in comparing the traces of sets $E^h_i$ defined on $\partial \Omega$ using $\Omega _\epsilon ^* = \cap (\Omega ^h_m)_\epsilon ^*$ with the traces of sets $E_i$ defined on $\partial \Omega$ using $\Omega _\epsilon ^* = \cap (\Omega _m)_\epsilon ^*$. The main difficulty is that the contact points do not appear in the same places. To address this, for each fixed h, we use $\left( \cap _m(\Omega ^h_m)_\epsilon ^*\right) \cap \Omega _\epsilon ^*$ when comparing the traces of $\phi _{E_i}$ and $\phi _{E_i^h}$.

For the traces of sets $E^h_i$ defined on ${\mathcal {B}}^h_k$ to be compared to the traces of sets $E_i$ defined on ${\mathcal {B}}_k$, we consider the linear transformation $T^h_k$ that maps ${\mathcal {B}}^h_k$ onto ${\mathcal {B}}_k$. This can also be used to map $({\mathcal {B}}_k^h)_{\epsilon }^{*}$ onto $({\mathcal {B}}_k)_{\epsilon }^{*}$, or at least onto a portion of $({\mathcal {B}}_k)_{\epsilon }^{*}$ that is sufficient for defining the trace of both $T^h_kE^h_i$ and $E_i$. This set is $T^h_k({\mathcal {B}}_k^h)_{\epsilon }^{*}\cap ({\mathcal {B}}_k)_{\epsilon }^{*}$. One might wish to remove the dependence on a specific h; however, if there is an infinite spin in our maps $T^h_k$, then it may be possible to have a countable dense set of contact points on $\partial {\mathcal {B}}_k$ and this would collapse our dimpled open neighborhood of the boundary on a compact set of positive measure. We do note that $T^h_k({\mathcal {B}}_k^h)_{\epsilon }^{*}\cap ({\mathcal {B}}_k)_{\epsilon }^{*} \rightarrow ({\mathcal {B}}_k)_{\epsilon }^{*}$ as $h\rightarrow \infty$ in the sense of characteristic functions converging in $L^1(\Omega )$ and that there are only a finite number of contact points on any given $T^h_k({\mathcal {B}}_k^h)_{\epsilon }^{*}\cap ({\mathcal {B}}_k)_{\epsilon }^{*}$.

We begin with a pair of preliminary inequalities:

Claim 2.4

For traces defined on the sets as above, we have

$$\begin{aligned} \sum ^3_{i=1} \beta _i\int _{\partial \Omega }\left( \phi _{E_i} - \phi _{E_i^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1} \le \sum _{j=1,3} |\beta _j - \beta _2|\int _{\partial \Omega }\left( \phi _{E_j} - \phi _{E_j^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1},\qquad \quad \end{aligned}$$

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and

$$\begin{aligned}&\sum ^3_{i=1} \tau _{ik}\int _{\partial {\mathcal {B}}_k}\left( \phi _{E_i} - \phi _{E_i^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1} \nonumber \\&\quad \le \sum _{j=1,3} |\tau _{p_k(j)k} - \tau _{p_k(2)k}|\int _{\partial {\mathcal {B}}_k}\left( \phi _{E_{p_k(j)}} - \phi _{E_{p_k(j)}^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1}. \end{aligned}$$

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Proof

Using $\sum \phi _{E_i} = 1$ and $\sum \phi _{E^h_i} = 1$, we have

$$\begin{aligned} 2\int _{\partial \Omega }\left( \phi _{E_1} - \phi _{E_1^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1} = \int _{\partial \Omega }\left( \left( \phi _{E_1} - \phi _{E_2} - \phi _{E_3} \right) - \left( \phi _{E^h_1} - \phi _{E^h_2} - \phi _{E_3^h}\right) \right) \, {\mathrm{d}}{\mathcal {H}}^{n-1}. \end{aligned}$$

Then, we multiply by $\beta _1$ and use the fact that $\beta _1\le \beta _2$ to find

$$\begin{aligned} 2\beta _1\int _{\partial \Omega }\left( \phi _{E_1} - \phi _{E_1^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1} \le \beta _2\int _{\partial \Omega }\left[ \left( \phi _{E_1} - \phi _{E^h_1} \right) - \left( \phi _{E_2} - \phi _{E^h_2} \right) - \left( \phi _{E_3} - \phi _{E_3^h}\right) \right] \, {\mathrm{d}}{\mathcal {H}}^{n-1}. \end{aligned}$$

Then, we add $\beta _3\int _{\partial \Omega }\left( \phi _{E_3} - \phi _{E_3^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1}$ to both sides and rearrange to find

$$\begin{aligned}&\sum ^3_{i=1} \beta _i\int _{\partial \Omega }\left( \phi _{E_i} - \phi _{E_i^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1} \le \\&\qquad \qquad (\beta _2 - \beta _1)\int _{\partial \Omega }\left( \phi _{E_1} - \phi _{E_1^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1} + (\beta _3 - \beta _2)\int _{\partial \Omega }\left( \phi _{E_3} - \phi _{E_3^h}\right) \, {\mathrm{d}}{\mathcal {H}}^{n-1}, \end{aligned}$$

establishing the first inequality. The second inequality is proved in the same manner. $\square$

Next, using our discussion on traces, Lemma 2.1, and our claim, we have

(25)

(26)

(27)

(28)

We treat the first terms of (28) in the following claim.

Claim 2.5

Let $\epsilon >0$ be a fixed number. As $h\rightarrow \infty$, we have

Proof

We first observe that for each $i = 1,2,3$, as $h\rightarrow \infty$, we have $\phi _{E_i^h} \rightarrow \phi _{E_i}$ in $L^1(\Omega )$. Combining this convergence with the fact that for each $i = 1,2,3$, we have , and this implies the expected fact that

Then, we will argue using the definition of the total variation by fixing a function where $||g||^0 \le 1$. With this function, and the above convergence, we have, for each $i = 1,2,3$,

Next, we observe that in a uniform manner, as each body ${\mathcal {B}}^h_k$ is a rigid motion of ${\mathcal {B}}_k$. Then, using the fact that , there exists $h_0\in {\mathbb {N}}$ large enough so that when $h\ge h_0$. Given such an $h\ge h_0$, it follows that we can use this g as an admissible function when defining for each $i = 1,2,3$. Taking a supremum over all with $||g||^0 \le 1$, we have

for each $i=1,2,3$, and when $h>h_0$. Then, combining the two quantities displayed above, we have

for each $i=1,2,3$. Finally, we take the supremum over all with $||g||^0 \le 1$ to complete the proof of the claim. $\square$

As $\phi _{E_j^h}\rightarrow \phi _{E_j}$ in $\L ^1(\Omega )$, we have

$$\begin{aligned} c|\beta _j - \beta _2|\int _{\Omega _\epsilon ^*}|\phi _{E_j} - \phi _{E_j^h}|\, {\mathrm{d}}x \rightarrow 0. \end{aligned}$$

Also, as $\phi _{T^h_kE_j^h}\rightarrow \phi _{E_j}$ in $\L ^1(\Omega )$, we have

Then, as $\epsilon \rightarrow 0$, both $|\Omega _\epsilon ^*|\rightarrow 0$ and ; thus, we have $\int _{\Omega _\epsilon ^*}|D\phi _{E_i}|\rightarrow 0$, ,

$$\begin{aligned} \sqrt{1+L^2}|\beta _j - \beta _2|\int _{\Omega _\epsilon ^*}|D\phi _{E_j}|\rightarrow 0, \end{aligned}$$

and

Next, we have the terms

$$\begin{aligned} -\alpha _2\int _{\Omega _\epsilon ^*} |D\phi _{E_2^h}| + \sum _{j=1,3}\left( \sqrt{1+L^2}|\beta _j - \beta _2| - \alpha _j\right) \int _{\Omega _\epsilon ^*}|D\phi _{E_j^h}|. \end{aligned}$$

(29)

This quantity is negative when $\sqrt{1+L^2}|\beta _j - \beta _2| - \alpha _j \le 0$, so we consider the case where $\alpha _1 < \sqrt{1+L^2}|\beta _1 - \beta _2|$. Using $\sum \phi _{E_j^h} = 1$ and $\beta _1\le \beta _2\le \beta _3$, we establish the following estimate for (29)

$$\begin{aligned}&\sum _{j=1,3}\left( \sqrt{1+L^2}|\beta _j - \beta _2| - \alpha _j\right) \int _{\Omega _\epsilon ^*}|D\phi _{E_j^h}| -\alpha _2\int _{\Omega _\epsilon ^*} |D\phi _{E_2^h}| \nonumber \\&\quad = \left( \sqrt{1+L^2}(\beta _2 - \beta _1) - \alpha _1\right) \int _{\Omega _\epsilon ^*}|D(1 -\phi _{E_2^h} - \phi _{E_3^h})| -\alpha _2\int _{\Omega _\epsilon ^*} |D\phi _{E_2^h}| \nonumber \\&\qquad + \left( \sqrt{1+L^2}(\beta _3 - \beta _2) - \alpha _3\right) \int _{\Omega _\epsilon ^*}|D\phi _{E_3^h}| \end{aligned}$$

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$$\begin{aligned}&\quad \le \left( \sqrt{1+L^2}(\beta _2 - \beta _1) - \alpha _1 - \alpha _2 \right) \int _{\Omega _\epsilon ^*}|D\phi _{E_2^h}| \nonumber \\&\qquad + \left( \sqrt{1+L^2}(\beta _3 - \beta _1) - \alpha _1 - \alpha _3 \right) \int _{\Omega _\epsilon ^*}|D\phi _{E_3^h}|. \end{aligned}$$

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Then, our assumption that $\alpha _i + \alpha _j \ge \sqrt{1+L^2}|\beta _i - \beta _j|$ implies that the end of our estimate is nonpositive. The case when $\alpha _3 < \sqrt{1+L^2}|\beta _3 - \beta _2|$ is slightly different. Our estimate is

$$\begin{aligned}&\sum _{j=1,3}\left( \sqrt{1+L^2}|\beta _j - \beta _2| - \alpha _j\right) \int _{\Omega _\epsilon ^*}|D\phi _{E_j^h}| -\alpha _2\int _{\Omega _\epsilon ^*} |D\phi _{E_2^h}| \nonumber \\&\quad \le \left( \sqrt{1+L^2}(\beta _2 - \beta _1) - \alpha _1 - \alpha _3 \right) \int _{\Omega _\epsilon ^*}|D\phi _{E_1^h}| \nonumber \\&\qquad + \left( \sqrt{1+L^2}(\beta _3 - \beta _2) - \alpha _2 - \alpha _3 \right) \int _{\Omega _\epsilon ^*}|D\phi _{E_2^h}| \nonumber \\&\quad \le \left( \sqrt{1+L^2}(\beta _3 - \beta _1) - \alpha _1 - \alpha _3 \right) \int _{\Omega _\epsilon ^*}|D\phi _{E_1^h}| \nonumber \\&\qquad + \left( \sqrt{1+L^2}(\beta _3 - \beta _2) - \alpha _2 - \alpha _3 \right) \int _{\Omega _\epsilon ^*}|D\phi _{E_2^h}|, \end{aligned}$$

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and the same conclusion holds. The remaining terms are

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and, upon splitting the second integral using , the above proof applies in the same manner. Therefore, ${\mathcal {A}}$ is lower semicontinuous. $\square$

Next, we include the gravity terms to treat the whole energy functional.

Lemma 2.6

Suppose that the hypotheses of Lemma 2.3 are satisfied. Then,

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Proof

By assumption, each ${\mathcal {B}}_k(c_k, R_k)$ is a rigid object, and by hypothesis, we have convergent sequences $(c_k^h, R_k^h) \rightarrow (c_k^0, R_k^0)$. Thus, each ${\mathcal {B}}_k(c_k^h, R_k^h)$ converges uniformly to the corresponding ${\mathcal {B}}_k(c_k^0, R_k^0)$, so our $\rho _{{\mathcal {B}}_k} g$ terms are continuous by the uniform limit theorem. For the $\rho _{E_i} g$ terms, we will use Fatou’s Lemma, which requires a nonnegative integrand. If $m := \min \{x_n: (x^\prime ,x_n)\in \Omega \} \ge 0$, then those terms are lower semicontinuous. Otherwise, we add quantities to the integrals resulting in

and then, we apply Fatou’s lemma and then subtract the terms with m to return to the needed form of ${\mathcal {F}}$. Finally, we note that we have already shown the lower semicontinuity of in Lemma 2.3. $\square$

Lastly, we have the main result.

Theorem 2.7

Let $\alpha _i, \rho _i, \rho _{{\mathcal {B}}_k} \in {\mathbb {R}}$ be such that $\rho _i \ge 0$, $\rho _{{\mathcal {B}}_k} \ge 0$, $\alpha _i \ge 0$, $\alpha _i + \alpha _j > 0$. We also require two conditions which include a Lipschitz constant L:

$$\begin{aligned} \alpha _i + \alpha _j \ge \sqrt{1+L^2}|\beta _i - \beta _j|, \end{aligned}$$

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and

$$\begin{aligned} \alpha _i + \alpha _j \ge \sqrt{1+L^2}|\tau _{ik} - \tau _{jk}|, \end{aligned}$$

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for each k. In addition suppose that we have a sequence and a configuration with . Then, there exists a such that for all ,

Proof

By Lemma 2.2, ${\mathcal {F}}$ is bounded from below, and we can form a minimizing sequence . We may assume that where $m_0$ is the value of ${\mathcal {F}}$ at the first element of the sequence. If this inequality does not hold for some elements, discard them and form a subsequence by relabeling. Therefore, $\phi _{E_i},\, i = 1, 2, 3$, are bounded in $BV(\Omega )$.

The values of each $R_k^h$ belong to a compact set, so each $\vert R_k^h\vert$ is bounded in ${\mathbb {R}}$ for all $h \in {\mathbb {N}}$. We also have bounds on each $c_k$, as each body satisfies ${\mathcal {B}}_k\subset \Omega$.

Therefore, our sequence is bounded in the sense that for all $h \in {\mathbb {N}}$. Since our sequence was a minimizing sequence, we can form a minimizing convergent subsequence, which we again call . By Lemma 2.6, ${\mathcal {F}}$ is lower semicontinuous with respect to this sequence; hence, the sequence converges in ${\mathcal {C}}$ to some , as desired. $\square$