Master 2
In the following, we propose a mathematical model for the energy of configurations of different immiscible fluids situated in a container , which is thought as an open subset of , . We consider fluids modeled by measurable sets 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 of functions with bounded variation on . 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 and for every , we define the total variation of on as
If and we write and say that has bounded variation in . Note that the space is contained in and for every we have
There are, however, functions in (for example characteristic functions of the smooth, open, bounded subsets of ).
One of the immediate properties of the total variation is that it is lower semicontinuous with respect to the topology, i.e. if is a sequence of functions which converges to in then
We can define a norm on in the following way:
and under this norm becomes a Banach space.
A useful property of is that for every the sets
are compact in the topology provided that is bounded and sufficiently regular such that the Rellich Theorem can be applied (for example has smooth boundary or Lipschitzcontinuous boundary).
There is an approximation result of bounded variation functions by smooth functions, but the approximation cannot be expected to take place in the norm, since then we know that . Consider . Then there exists a sequence such that
and
As a consequence we obtain that the sets
are compact in for all constants .
For a fixed open subset of the map
defined for any open subset of is the trace on the open sets of a uniquely determined Borel measure on ; we shall denote by the value of this measure on a measurable subset of .
We denote by the dimensional Hausdorff measure and the Lebesgue measure on . If is any measurable subset of we denote by the characteristic function of and for every open subset of we let
If we say that has finite perimeter in . It can be proved that with equality if, for instance, is a Lipschitz continuous hypersurface. Here are a few properties of the perimeter:
 If then with equality if .
 with equality if .
 implies and implies .
If is an open subset of and then the function
is Lebesgue measurable on and the FlemingRishel coarea formula holds:
For every set with finite perimeter it is possible to construct a subset called the reduced boundary of , such that
One important property of 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 functions. Assume is open, bounded and has Lipschitz boundary. Then there exists a bounded linear mapping
such that
for all and .
The function , which is uniquely defined up to sets of measure zero, is called the trace of on . We also have for a.e.
which implies that
For more details see [1] Chapter 5.
We now desire to give a formulation for the energy of a configuration of fluids, taking into account the interfacial tension between the fluids, the gravity and the contact with the walls of the container. Suppose the container is a bounded open subset of , usually with some regularity assumptions like is Lipschitz continuous or . Consider the fluids modeled by the measurable sets such that and , equalities which hold up to a measurable set. Each has a prescribed volume and a given density such that . We know the interfacial tensions at the interface between and , and the wetting coefficient of with the wall of the container . We also assume that the sets have finite perimeter in , which allows us to speak of their reduced boundary .
Now we are able to define a formula for the energy of the system
where the last term is the gravitational potential energy.
Our goal is to prove that under certain conditions on , and the regularity of the problem
has a solution in , where and
The equalities are considered modulo a set of measure zero, and .
[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,

March 1, 2013 at 11:59 amMaster 4  Beni Bogoşel's blog

March 1, 2013 at 12:46 pmMaster 5  Beni Bogoşel's blog