## Master 2

In the following, we propose a mathematical model for the energy of configurations of ${n}$ different immiscible fluids situated in a container ${\Omega}$, which is thought as an open subset of ${\Bbb{R}^N}$, ${N \geq 2}$. We consider fluids modeled by measurable sets ${E \subset \Omega}$ and since we are going to study functionals which deal with the surface tension energies we will need a good framework for studying the perimeters of such sets. For this we consider the space ${BV(\Omega)}$ of functions with bounded variation on ${\Omega}$. Some of the standard references for this subject are the books of Evans, Gariepy [1] and E. Giusti [2]. We will state below some of the results we will be using in the sequel, whose proofs can be found in the given references.

For any open subset ${\Omega \subset \Bbb{R}^N,\ N \geq 2}$ and for every ${u \in L^1_{loc}(\Omega)}$, we define the total variation of ${u}$ on ${\Omega}$ as

$\displaystyle \int_\Omega |D u|=\sup\left\{ \int_\Omega u(x) \text{div} g(x) dx : g \in C_0^\infty (\Omega ;\Bbb{R}^N),\ |g| \leq 1 \right\}$

If ${\int_\Omega |u| <\infty}$ and ${\int_\Omega |D u|<\infty}$ we write ${u \in BV(\Omega)}$ and say that ${u}$ has bounded variation in ${\Omega}$. Note that the space ${W^{1,1}(\Omega)}$ is contained in ${BV(\Omega)}$ and for every ${u \in W^{1,1}(\Omega)}$ we have

$\displaystyle \int_\Omega |D u|=\int_\Omega |D u(x)|dx.$

There are, however, functions in ${BV(\Omega) \setminus W^{1,1}(\Omega)}$ (for example characteristic functions of the smooth, open, bounded subsets of ${\Omega}$).

One of the immediate properties of the total variation is that it is lower semicontinuous with respect to the ${L^1(\Omega)}$ topology, i.e. if ${(u_n) \subset L^1(\Omega)}$ is a sequence of functions which converges to ${u}$ in ${L^1(\Omega)}$ then

$\displaystyle \int_\Omega |D u| \leq \liminf_{n \rightarrow \infty} \int_\Omega |D u_n|.$

We can define a norm on ${BV(\Omega)}$ in the following way:

$\displaystyle \|u\|_{BV}=\|u\|_{L^1}+\int_\Omega |D u|,$

and under this norm ${BV(\Omega)}$ becomes a Banach space.

A useful property of ${BV(\Omega)}$ is that for every ${c>0}$ the sets

$\displaystyle \{ u \in BV(\Omega): \int_\Omega |u|dx+ \int_\Omega |D u| \leq c \}$

are compact in the ${L^1(\Omega)}$ topology provided that ${\Omega}$ is bounded and sufficiently regular such that the Rellich Theorem can be applied (for example ${\Omega}$ has smooth boundary or Lipschitz-continuous boundary).

There is an approximation result of bounded variation functions by smooth functions, but the approximation cannot be expected to take place in the ${BV}$ norm, since then we know that ${BV(\Omega)\neq W^{1,1}(\Omega)}$. Consider ${u \in BV(\Omega)}$. Then there exists a sequence ${(u_n) \subset C^\infty(\Omega)}$ such that

$\displaystyle \lim_{n \rightarrow \infty} \int_\Omega |u_n-u|dx=0$

and

$\displaystyle \lim_{n \rightarrow \infty} \int_\Omega |D u_n|dx = \int_\Omega |D u|.$

As a consequence we obtain that the sets

$\displaystyle \left\{ u \in BV(\Omega) : \int_\Omega u=m,\ \int_\Omega |D u|\leq c \right\}$

are compact in ${L^1(\Omega)}$ for all constants ${m \in \Bbb{R},c \in \Bbb{R}_+}$.

For a fixed open subset ${\Omega}$ of ${\Bbb{R}^N}$ the map

$\displaystyle A \mapsto \int_A |D u|$

defined for any open subset ${A}$ of ${\Omega}$ is the trace on the open sets of a uniquely determined Borel measure on ${\Omega}$; we shall denote by ${\int_E |D u|}$ the value of this measure on a measurable subset ${E}$ of ${\Omega}$.

We denote by ${\mathcal{H}^{N-1}}$ the ${N-1}$-dimensional Hausdorff measure and ${|\cdot |}$ the Lebesgue measure on ${\Bbb{R}^N}$. If ${E}$ is any measurable subset of ${\Bbb{R}^N}$ we denote by ${\chi_E}$ the characteristic function of ${E}$ and for every open subset ${\Omega}$ of ${\Bbb{R}^N}$ we let

$\displaystyle \text{Per}_\Omega(E)=\int_\Omega |\chi_E|.$

If ${\text{Per}_\Omega(E)<\infty}$ we say that ${E}$ has finite perimeter in ${\Omega}$. It can be proved that ${P_\Omega(E)\leq \mathcal{H}^{N-1}(\partial E \cap \Omega)}$ with equality if, for instance, ${\partial E \cap \Omega}$ is a Lipschitz continuous hypersurface. Here are a few properties of the perimeter:

• If ${\Omega' \subset \Omega}$ then ${\text{Per}_{\Omega'}(E) \leq \text{Per}_\Omega(E)}$ with equality if ${E \subset \subset \Omega'}$.
• ${\text{Per}_\Omega(E_1 \cup E_2) \leq \text{Per}_\Omega(E_1)+\text{Per}_\Omega(E_2)}$ with equality if ${d(E_1,E_2)>0}$.
• ${|E|=0}$ implies ${\text{Per}_\Omega(E)=0}$ and ${|E_1 \Delta E_2|=0}$ implies ${\text{Per}_\Omega(E_1)=\text{Per}_\Omega(E_2)}$.

If ${\Omega}$ is an open subset of ${\Bbb{R}^N}$ and ${u \in BV(\Bbb{R}^N)}$ then the function

$\displaystyle t \mapsto P_\Omega(\{x \in \Bbb{R} : u(x)>t \})$

is Lebesgue measurable on ${\Bbb{R}}$ and the Fleming-Rishel coarea formula holds:

$\displaystyle \int_\Omega |Du| =\int_{-\infty}^\infty P_\Omega (\{x \in \Bbb{R}^N : u(x)>t \})dt.$

For every set ${S}$ with finite perimeter it is possible to construct a subset ${\partial^* S \subset \partial S}$ called the reduced boundary of ${S}$, such that

$\displaystyle P_\Omega(S)=\mathcal{H}^{N-1}(\partial^*S \cap \Omega).$

One important property of ${BV(\Omega)}$ is that when the boundary is sufficiently regular there is a well defined trace of a bounded variation function on the boundary which agrees with the definition for ${W^{1,1}(\Omega)}$ functions. Assume ${\Omega}$ is open, bounded and has Lipschitz boundary. Then there exists a bounded linear mapping

$\displaystyle T : BV(\Omega) \rightarrow L^1(\partial \Omega ; \mathcal{H}^{N-1})$

such that

$\displaystyle \int_\Omega f \text{div} \varphi dx = -\int_\Omega \varphi\cdot d |Df|+\int_{\partial \Omega}(\varphi \cdot \nu)Tf d\mathcal{H}^{N-1}$

for all ${f \in BV(\Omega)}$ and ${\varphi \in C^1(\Bbb{R}^N;\Bbb{R}^N)}$.

The function ${Tf}$, which is uniquely defined up to sets of ${\mathcal{H}^{N-1}}$ measure zero, is called the trace of ${f}$ on ${\partial \Omega}$. We also have for ${\mathcal{H}^{N-1}}$ a.e. ${x \in \partial \Omega}$

$\displaystyle \lim_{r \rightarrow 0} \frac{1}{|B(x,r) \cap \Omega|} \int_{B(x,r)\cap \Omega} |f(y)-Tf(x)|dy=0,$

which implies that

$\displaystyle Tf(x)=\lim_{r \rightarrow 0} \frac{1}{|B(x,r) \cap \Omega|} \int_{B(x,r) \cap \Omega} f(y)dy.$

For more details see [1] Chapter 5.

We now desire to give a formulation for the energy of a configuration of ${n}$ fluids, taking into account the interfacial tension between the fluids, the gravity and the contact with the walls of the container. Suppose the container ${\Omega}$ is a bounded open subset of ${\Bbb{R}^N}$, usually with some regularity assumptions like ${\partial \Omega}$ is Lipschitz continuous or ${C^1}$. Consider the fluids modeled by the measurable sets ${E_1,...,E_n \subset \Omega}$ such that ${E_1 \cup...\cup E_n=\Omega}$ and ${E_i \cap E_j=\emptyset,\forall 1 \leq i < j \leq n}$, equalities which hold up to a measurable set. Each ${E_i}$ has a prescribed volume ${c_i>0}$ and a given density ${\rho_i>0}$ such that ${c_1+...+c_n=|\Omega|}$. We know the interfacial tensions ${\sigma_{ij}>0}$ at the interface between ${E_i}$ and ${E_j}$, and the wetting coefficient ${\beta_i}$ of ${E_i}$ with the wall of the container ${\Omega}$. We also assume that the sets ${E_i}$ have finite perimeter in ${\Omega}$, which allows us to speak of their reduced boundary ${\partial^* E_i}$.

Now we are able to define a formula for the energy of the system

$\displaystyle \mathcal{E}(E_1,...,E_n)=\sum_{i,j=1, i \neq j}^n \sigma_{ij}\mathcal{H}^{N-1} (\partial^* E_i \cap \partial^* E_j \cap \Omega)+\sum_{i=1}^n\beta_i \mathcal{H}^{N-1}(\partial^* E_i \cap \partial \Omega)+$

$\displaystyle + \sum_{i=1}^n g\rho_i \int_\Omega x_N\chi_{E_i}dx$

where the last term is the gravitational potential energy.

Our goal is to prove that under certain conditions on ${\sigma_{ij}}$, ${\beta_i}$ and the regularity of ${\partial\Omega}$ the problem

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

has a solution in ${\mathcal{K}}$, where ${E=(E_1,...,E_n)}$ and

$\displaystyle \mathcal{K}=\{ (E_i)_{i=1}^n : \bigcup_{i=1}^n E_i=\Omega,\ |E_i|=c_i, E_i \cap E_j=\emptyset \}$

The equalities are considered modulo a set of measure zero, and ${c_1+...+c_n=|\Omega|}$.

[1] Giusti, Enrico, Minimal surfaces and functions of bounded variation

[2] Evans, Lawrence C. and Gariepy, Ronald F., Measure theory and fine properties of functions,