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,

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  1. March 1, 2013 at 11:59 am
  2. March 1, 2013 at 12:46 pm

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