Home > shape optimization > Master part 3

## Master part 3

This is the third part of my series of posts regarding my master thesis. Here I follow the article of Massari [1] regarding the formulation of the problem and the proof of the existence result in the case of two fluids. Although the ideas are the same, the proof presented here is different from the one given in the referred article.

Suppose we have two fluids ${E_1,E_2}$, with the considerations above. Then the energy functional has the following form

$\displaystyle \mathcal{E}(E_1,E_2)=\sigma_{12}\mathcal{H}^2 (\partial^* E_1 \cap \partial^* E_2 \cap \Omega)+\sum_{i=1}^2\beta_i \mathcal{H}^2(\partial^* E_i \cap \partial \Omega)+\sum_{i=1}^2 g\rho_i \int_\Omega x_N\chi_{E_i}dx, \ \ \ \ \ (1)$

but since ${\chi_{E_1}+\chi_{E_2}=1}$ almost everywhere on ${\Omega}$, we see that the position of ${E_2}$ is uniquely determined by the position of ${E_1}$, so it is obvious that the problem

$\displaystyle \min_{(E_1,E_2) \in \mathcal{K}} \mathcal{E}(E_1,E_2)$

has a solution if and only if the problem

$\displaystyle \min_{E\subset \Omega,|E|=c} \mathcal{F}(E)$

has a solution, where

$\displaystyle \mathcal{F}(E)=\gamma \text{Per}_\Omega(E)+\beta\mathcal{H}^{N-1}(\partial^*E \cap \partial \Omega)+g\rho \int_\Omega x_N\chi_E dx. \ \ \ \ \ (2)$

This formulation can be proven to be equivalent to the initial one, by replacing ${E_2=\Omega\setminus E_1}$ in ${\mathcal{E}(E_1,E_2)}$. Notice that the term ${\mathcal{H}^2 (\partial^* E_1 \cap \partial^* E_2\cap \Omega)}$ is in fact equal to ${\text{Per}_\Omega(E_1)=\text{Per}_\Omega(E_2)}$, since we have only two fluids, and the measure interface between them, which is inside ${\Omega}$, is equal to their relative perimeter with respect to ${\Omega}$. Also, notice that ${\mathcal{H}^2(\partial^* E_1 \cap \partial\Omega)+\mathcal{H}^2(\partial^* E_2 \cap \partial \Omega)=\text{Per}(\Omega)}$. Then we have

$\displaystyle \mathcal{E}(E_1,E_2)=\sigma_{12}\text{Per}_\Omega(E_1)+(\beta_1-\beta_2)\mathcal{H}^2(\partial^*E_1 \cap \partial \Omega)+g(\rho_1-\rho_2) \int_\Omega x_N\chi_{E_1} dx+C$

where ${C=\beta_2 \text{Per}(\Omega)+g\rho_2 \int_{\Omega}x_N dx}$. Notice that this has the same form as (2) with ${\gamma=\sigma_{12},\ \beta=\beta_1-\beta_2}$ and ${\rho=\rho_1-\rho_2}$ except for the constant ${C}$, which does not affect the existence of the minimizer.

There are a few simple conditions which are needed for the existence of the minimizer for (2). First note that the term ${\beta \mathcal{H}^2(\partial^* E\cap \partial \Omega)}$ is bounded by ${|\beta|\text{Per}(\Omega)}$ and the last term is bounded by ${|g|\rho \int_\Omega \max\limits_\Omega |x_N| dx}$. If we have ${\gamma<0}$ then ${\mathcal{F}}$ is not bounded from below, and the minimizer does not exist. Therefore, a necessary condition for the existence of a minimum is ${\gamma \geq 0}$. This is a reasonable condition from a physical point of view, because it simply says that ${\sigma_{12}\geq 0}$, which is true. In fact ${\sigma_{12}>0}$, or else the two fluids would mix, which is not the case here. Hence, in the following we will use the condition ${\gamma>0}$.

Therefore, if ${\gamma > 0}$ we know that ${\mathcal{F}}$ is bounded from below. Suppose ${E^h}$ is a minimizing sequence. Then ${\mathcal{F}(E^h)}$ is bounded above by a constant ${K>0}$, and therefore

$\displaystyle \text{Per}(E^h) \leq \frac{1}{\gamma}\left(K+|\beta|\text{Per}(\Omega)+|g|\rho \int_\Omega \max_\Omega |x_N| dx\right) \ \ \ \ \ (3)$

Therefore ${\text{Per}(E^h)}$ is bounded and since ${|E^h|=c}$, by the compactness property it follows that the sequence ${\chi_{E^h}}$ has a convergent subsequence in the ${L^1(\Omega)}$ topology. Since the perimeter is lower semicontinuous for the ${L^1(\Omega)}$ topology, it follows that the limit point of the convergent subsequence of ${(E^h)}$ is also a set with finite perimeter. If we could prove that ${\mathcal{F}}$ is lower semicontinuous with respect to the ${L^1(\Omega)}$ topology, then the existence of a minimizer would be proved.

First note that if ${|\beta|> \gamma}$ then the functional ${\mathcal{F}}$ is not lower semicontinuous.

Example 1 Take ${\Omega=\{ (x,y,z) : x^2+y^2+z^2<1 \text{ for }z \leq 0,\ x^2+y^2 < 1, z \in (0,2) \text{ and }x^2+y^2+(z-2)^2 < 1\text{ for }z \geq 2 \}}$ (a cylinder with two halfspheres glued at its ends). Note that ${\Omega}$ has ${C^1}$-boundary and the interior sphere property: there exists ${\rho>0}$ such that for any ${x \in \Omega}$ there exists a ball of radius ${\rho}$ such that ${x \in B_\rho \subset \Omega}$.

Take ${S}$ the open unit ball centered in origin, and denote ${S_\varepsilon}$ the translation of ${S}$ with vector ${(0,0,\varepsilon)}$, where ${\varepsilon >0}$.

Note that for ${\varepsilon<1}$ we have ${S_\varepsilon \subset \Omega}$ and ${S_\varepsilon}$ converges to ${S=S_0}$ in ${L^1(\Omega)}$ as ${\varepsilon \rightarrow 0}$. First, let’s notice that for ${\varepsilon >0}$ ${S_\varepsilon \subset \Omega}$ and ${\mathcal{F}(S_\varepsilon)=\gamma\text{Per}_\Omega(S_\varepsilon)+\mathcal{G}(S_\varepsilon)}$, where ${\mathcal{G}}$ is the gravity term, and it is ${L^1(\Omega)}$ continuous.

On the other hand ${ \displaystyle \mathcal{F}(S)=\frac{\gamma}{2}\text{Per}(S)+\frac{\beta}{2}\text{Per}(S)+\mathcal{G}(S)}$, and therefore ${\displaystyle \lim_{\varepsilon \rightarrow 0}[\mathcal{F}(S)-\mathcal{F}(S_\varepsilon)]=\frac{\beta-\gamma}{2}\text{Per}(S)}$. This limit is strictly positive for ${\beta>\gamma}$ and it follows that ${\mathcal{F}}$ is not necessarily lower semicontinuous if ${\beta>\gamma}$.

Example 2 Consider the cube ${C=[-1,1]^3}$ and ${S}$ the unit open ball. Define ${\Omega=C+\frac{1}{10}S=\{a+b: a\in C,b \in \frac{1}{10}S\}}$. Then ${\Omega}$ is open and has ${C^1}$ boundary and the interior sphere property.

Let’s consider now sets ${E \subset \Omega}$ such that ${|E|=c \in (0,|\Omega|)}$ and ${E}$ is the complement in ${\Omega}$ of a compact subset ${K \subset \Omega}$. In particular, this means that ${\partial \Omega \subset \partial^* E}$, so the functional ${\mathcal{F}}$ has the following form

$\displaystyle \mathcal{F}(E)=\gamma \text{Per}_\Omega(E)+ \beta \text{Per}(\Omega)+\mathcal{G}(E),$

where ${\mathcal{G}}$ is the gravity term, and it is continuous. Consider sets of the form ${\Omega_r=aC+rS}$ such that ${|aC+rS|=c}$. The volume of ${aC+rS}$ can explicitly be calculated in terms of ${a,r}$, because ${aC+rS}$ can be split in parts which form a cube of side ${a}$, six boxes of size ${a \times a \times r}$, three cylinders with radius ${r}$ and height ${a}$ and a ball of radius ${r}$,

$\displaystyle |aC+rS|=a^3+6a^2r+3\pi ar^2+\frac{4\pi r^3}{3}$

note that for every ${r}$ such that ${4\pi r^3/3 we can find ${a>0}$ such that ${|aC+rS|=c}$. Denote ${a(r)}$ the positive solution of this equation and note that as ${4\pi r^3/3 \rightarrow c}$ we must have ${a(r) \rightarrow 0}$. Therefore, the diameter of the figure ${\Omega_r=|a(r)C+r(S)|}$ in the direction of the axes ${Ox,Oy,Oz}$, which is equal to ${2r+a(r)}$ tends to ${2r_{max}}$ which is the diameter of the sphere of volume ${c}$.

Pick ${c}$ such that the sphere of volume ${c}$ centered at the origin doesn’t fit inside ${\Omega}$ (pick the diameter of the sphere equal to ${2+2/10+\varepsilon}$ where ${\varepsilon>0}$ is small enough. By the continuity of ${r \mapsto 2r+a(r)}$ we can deduce that there is an ${r_0}$ such that ${2r_0+a(r_0)}$ is equal to ${2+2/10}$, and ${a(r_0)\neq 0}$ since the ball of radius ${r_0}$ centered at the origin does not fit in ${\Omega}$. Moreover, by implicit differentiation, we see that ${2r+a(r)}$ is increasing. Consider now the limit

$\displaystyle \lim_{r \rightarrow r_0,r

$\displaystyle =\lim_{r \rightarrow r_0}\big[\gamma \text{Per}_\Omega(\Omega_{r_0})-\gamma \text{Per}_\Omega(\Omega_r)+ \beta(\mathcal{H}^{2}(\partial \Omega \setminus \partial\Omega_{r_0}) -\text{Per}(\Omega))+$

$\displaystyle +\mathcal{G}(\Omega_{r_0})-\mathcal{G}(\Omega_r)\big]= (-\gamma-\beta)6a^2(r_0)$

Notice now, that if ${\beta<-\gamma}$ then ${-\gamma-\beta>0}$ and the result of the above limit is strictly positive. This proves that ${\mathcal{F}}$ is not necessarily lower semicontinuous if ${\beta <-\gamma}$.

In the following, for a function ${u \in BV(\Omega)}$ we denote by ${\tilde {u}}$ its trace on ${\partial \Omega}$.

Definition 1 Suppose ${\Omega \subset \Bbb{R}^N}$ is an open set with Lipshitz boundary and ${\mathcal{H}^{N-1}(\Omega)<\infty}$. For every ${x \in \partial \Omega}$ we define$\displaystyle q(x)=\limsup_{\rho \rightarrow 0} \left\{ \frac{\int_{\partial \Omega} \tilde {\chi_A} d\mathcal{H}^{N-1}}{\int_\Omega |D \chi_A|} : A \subset \Omega \cap B(x,\rho),\ |A|>0,\ \text{Per}_\Omega(A)<\infty \right\}$

and denote ${Q_\Omega =\sup\limits_{x \in \partial \Omega} q(x)}$.

As noted in [2] if ${\Omega}$ has boundary of class ${C^1}$ then ${Q_\Omega=1}$ and for open sets with Lipschitz boundary we have ${Q_\Omega=\sqrt{1+L^2}}$ where ${L}$ is the Lipschitz constant of ${\partial \Omega}$. The following trace inequality is proved in [2].

Proposition 2 Suppose ${\Omega \subset \Bbb{R}^N}$ is a bounded open set with Lipschitz boundary and ${\mathcal{H}^{N-1}(\partial\Omega)<\infty}$. Then for every ${\varepsilon >0}$ there exists a constant ${c=c(\Omega,\varepsilon)}$ such that for every ${u \in BV(\Omega)}$ we have$\displaystyle \int_{\partial \Omega}|\tilde {u}|d \mathcal{H}^{N-1} \leq (Q_\Omega+\varepsilon) \int_\Omega |D u|+c(\Omega,\varepsilon) \int_\Omega |u|dx.$

With the aid of the above result we will prove an inequality which is of great importance in the proofs of the theorems in this chapter.

Theorem 3 Suppose ${\Omega \subset \Bbb{R}^N}$ is an open set with Lipschitz boundary, ${\varepsilon,\rho > 0}$ and ${Q_\Omega}$ as in Definition 1. Define ${\Omega_\rho=\{x \in \Omega : d(x,\partial \Omega)<\rho\}}$. Then for every ${\rho>0}$ there exists a constant ${k}$ such that for every ${u \in BV(\Omega)}$ we have$\displaystyle \int_{\partial \Omega} |\tilde {u}| d\mathcal{H}^{N-1} \leq (Q_\Omega +\varepsilon)\int_{\Omega_\rho} |D u|+k\int_{\Omega_\rho} |u|dx. \ \ \ \ \ (4)$

Proof: We will use Proposition 2 for the function ${v=(1-\chi_\rho)u}$ where ${\chi_\rho \in C_0^1 (\Omega),\ 0\leq \chi_\rho \leq 1,\ \chi_\rho(x)=1}$ if ${d(x,\partial \Omega)\geq \rho}$ and ${|D \chi_\rho|\leq 2/\rho}$.

We have

$\displaystyle \begin{array}{rcl} \int_{\partial \Omega}|\tilde {u}|d\mathcal{H}^{N-1} & = \int_{\partial \Omega}|\tilde {v}|d\mathcal{H}^{N-1} \leq \\ & \leq (Q+\varepsilon)\int_\Omega |D (1-\chi_\delta)u|+c(\Omega,\varepsilon)\int_\Omega |(1-\chi_\delta)u|dx \leq \\ & \leq (Q+\varepsilon) \int_{\Omega_\rho} |D u|+\left(\frac{2(Q+\varepsilon)}{\rho}+c(\Omega,\varepsilon)\right)\int_{\Omega_\rho}|u|dx \end{array}$

and note that we can take ${k=\displaystyle \frac{2(Q+\varepsilon)}{\rho}+c(\Omega,\varepsilon)}$. \hfill ${\square}$

Italo Tamanini proves in [3] a similar inequality for open sets ${\Omega}$ with the interior sphere condition, i.e. there exists some ${r>0}$ such that for any ${x \in \Omega}$ there exists a ball of radius ${r}$ such that ${x \in B_r \subset\Omega}$. Equivalently, the curvature of ${\partial \Omega}$ is bounded from above. This inequality states that for every set with finite perimeter ${E \subset \Omega}$ we have

$\displaystyle \mathcal{H}^{N-1}(\partial^* E \cap \partial \Omega) \leq \text{Per}_{\Omega_\rho}(E)+c |E\cap \Omega_\rho|,$

where ${c}$ is a constant which depends on ${\rho}$ and ${\Omega}$. This is a stronger inequality than (4), but its proof uses in an essential way the interior sphere property of ${\Omega}$, and cannot be directly generalized to sets with Lipschitz boundary. This inequality is used in [1] in the proof of the existence result, but here we will use the inequality (4), as the proof still works with this weaker inequality.

Theorem 4 Suppose ${\Omega}$ is a bounded open set with ${C^1}$ boundary. If ${\gamma>0}$ and ${\gamma \geq |\beta|}$ then the functional ${\mathcal{F}}$ is lower semicontinuous for the ${L^1}$ convergence of characteristic functions and therefore the problem (2) has a solution.

Proof: Let’s consider cases ${\beta=\pm\gamma}$ separately. If ${\beta=\gamma}$ then the perimeter and boundary terms give ${\gamma\text{Per}(E)}$ which is lower semicontinuous. If ${\beta=-\gamma}$ then the perimeter and boundary terms differ by a constant of ${\gamma \text{Per}(\Omega \setminus E)}$ which is also lower semicontinuous. Therefore these cases do not pose any problem, and from now on we suppose that ${\gamma> |\beta|}$.

Consider ${(E^h)}$ a sequence of sets of finite perimeter such that ${E^h \rightarrow E}$ in the ${L^1(\Omega)}$ convergence. Consider ${\varepsilon,\rho >0}$ small enough and note that for ${\Omega}$ considered in the hypothesis we have ${Q_\Omega=1}$. We denote with ${\mathcal{G}(E)}$ the gravity term associated to ${E}$ and note that ${\mathcal{G}}$ is continuous with respect to the ${L^1(\Omega)}$ convergence. Denote ${\mathcal{F}'=\mathcal{F}-\mathcal{G}}$. Then applying inequality (4) we have

$\displaystyle \begin{array}{rcl} \mathcal{F}'(E)-\mathcal{F}'(E^h) &= \gamma \left( \int_\Omega |D \chi_E|-\int_\Omega |D \chi_{E^h}| \right)+\beta \int_{\partial \Omega} (\tilde {\chi_E}-\tilde {\chi_{E^h}})d\mathcal{H}^{N-1}+\\ & \leq \gamma \left( \int_\Omega |D \chi_E|-\int_\Omega |D \chi_{E^h}| \right)+|\beta|\int_{\partial \Omega} |\tilde {\chi_E}-\tilde {\chi_{E^h}}|d\mathcal{H}^{N-1}\leq \\ & \leq \gamma \left( \int_\Omega |D \chi_E|-\int_\Omega |D \chi_{E^h}| \right)+|\beta| (1+\varepsilon) \int_{\Omega_\rho} |D \chi_E|+\\ &+|\beta|(1+\varepsilon) \int_{\Omega_\rho} |D \chi_E^h|+k|\beta|\int_\Omega |\chi_E-\chi_{E^h}|dx = \\ & = \gamma \left( \int_{\Omega \setminus \Omega_\rho} |D \chi_E|-\int_{\Omega \setminus \Omega_\rho} |D \chi_{E^h}| \right)+\\ &+(\gamma+|\beta|(1+\varepsilon))\int_{\Omega_\rho}|D \chi_E|+(|\beta|(1+\varepsilon)-\gamma)\int_{\Omega_\rho}|D \chi_E^h|+\\ &+ k|\beta| \int_\Omega |\chi_E-\chi_{E^h}|dx \end{array}$

By choosing ${\varepsilon}$ small enough such that ${|\beta|(1+\varepsilon)<\gamma}$, using the fact that ${\mathcal{G}}$ is continuous and the total variation is lower semicontinuous we obtain that

$\displaystyle \limsup_{h \rightarrow \infty} \mathcal{F}(E)-\mathcal{F}(E^h)\leq (\gamma+|\beta|(1+\varepsilon))\int_{\Omega_\rho}|D \chi_E|.$

Since ${\rho>0}$ is arbitrary small, we can deduce that

$\displaystyle \limsup_{h \rightarrow \infty} \mathcal{F}(E)-\mathcal{F}(E^h)\leq 0$

which means that ${\mathcal{F}}$ is lower semicontinuous for the ${L^1(\Omega)}$ convergence of characteristic functions.

The boundedness condition (3) together with the compactness property now lead to the existence of a minimizer for problem (2). \hfill ${\square}$

Since the fact that ${\Omega}$ has ${C^1}$ boundary is a bit restricting (for examples, squares, rectangles, cubes and cylinders are not included in this class), we can state another variant of the result for domains with Lipschitz continuous boundary. Note that for ${\Omega}$ with Lipschitz continuous boundary we have ${Q_\Omega=\sqrt{1+L^2}}$ where ${L}$ is the Lipschitz constant of ${\partial \Omega}$. It is easy to see that if we impose the stronger hypothesis ${\gamma > \sqrt{1+L^2}\beta}$ then the proof of the next result works exactly as the proof of the above theorem.

Theorem 5 Suppose ${\Omega}$ is a bounded open set with Lipschitz continuous boundary. If ${\gamma>0}$ and ${\gamma > \sqrt{1+L^2}|\beta|}$ then the functional ${\mathcal{F}}$ is lower semicontinuous for the ${L^1}$ convergence of characteristic functions and therefore the problem (2) has a solution.

[1] Massari, U., The parametric problem of capillarity: The case of two and three fluids

[2] Anzellotti, G. and Giaquinta, M., BV functions and traces

[3] Tamanini, Italo, Il problema della capillarita su domini non regolari