Master 4
(If you are interested check the previous posts: Part 1, Part 2, Part 3)
We consider now the case where we have three fluids in the container . Then the energy functional has the form
where
Since we have
we can rewrite the energy functional in the following form
where satisfy the relations for . Indeed, we can calculate that
and by the physical considerations on the interfacial tensions it is reasonable to assume that . In fact, the condition is a necessary condition for the existence of a lower bound for the functional .
Remark 1 We cannot obtain a formulation similar to (1) for a number of four or more fluids, because the system of equations
is overdetermined, and has solution if and only if the following obvious condition holds on
for all indexes which are pairwise distinct. These conditions are not physically justified. Take for example the values of the interfacial tensions (measured in dynes) given in the Smithsonian Physical Tables for air, water, olive oil and mercury.
The lower bound for the energy functional can be deduced from the general form of the energy if , because then we trivially have
Of course, the condition is necessary, because otherwise, we can make the boundary between and to be very rough, making and therefore .
This means that one necessary condition for the existence of the minimum is for all . If one is zero, then the fluid can spread freely in , because the cost of the boundary of and becomes zero. This contradicts the immiscibility of the fluids and . Therefore, regarding this fact, and the physical considerations, we will assume that for all .
Let’s now turn to the compactness property. Denote . Then we have the following inequality:
If we choose a minimizing sequence for , then is bounded, and by the inequality above it follows that is bounded. Since this happens for every and , by the compactness property and a diagonal argument it follows that has a convergent subsequence in topology to some element such that .
To be able to prove the lower semicontinuity of the functional in the topology, in the case when has smooth boundary, we have the necessary condition , a condition which is similar with the condition we had in the case of two fluids. If these conditions do not hold, we have the following counterexample.
Example 1 In this example we suppose that and we will see that the functional is not lower semicontinuous. This example can be extended to more dimensions, or to multiple fluids to see that the condition is really necessary for the lower semicontinuity of the energy functional. The main idea is that if if we replace the contact region of with the container by a thin layer of then we get a strictly lower energy value.
Consider the unit square and define (this is the fixed position of ). Consider and (see the figure).
Define (we take a thin strip of and pass it to ). Notice that all terms concerning are constant, so we just look at the terms containing .
On the other hand
which proves that
and is not lower semicontinuous.
We are now ready to state the main result of this section.
Theorem 1 Consider an open set with boundary. If and for every with then the functional
is lower semicontinuous with respect to convergence of characteristic functions.
Proof: First let’s notice that the gravity term is continuous, and this reduces the problem to proving that
is lower semicontinuous.
Pick and consider such that in the sense of convergence of characteristic functions. Without loss of generality we can assume that . We notice that almost everywhere in . Therefore, we can replace and in terms of in the trace term from the expression of . We also take sufficiently small and use the trace inequality. Therefore we have
Denote
If we prove that then we are done, since
and the right hand side term tends to zero as . So let’s prove that . If then we are done. If , we obtain
for chosen small enough.
If then a similar inequality takes place. It is easy to see that if both and If hold then we contradict the hypothesis for small enough.
Therefore the functional is lower semicontinuous, and the theorem is proved.\hfill
As in the case of two fluids, we can formulate a similar result for with Lipschitz boundary by strengthening the inequalities between and such that the trace inequality for Lipschitz continuous domains can be applied.
Theorem 2 Consider an open set with Lipschitz continuous boundary. If and for every with ( is the Lipschitz constant of ) then the functional
is lower semicontinuous with respect to convergence of characteristic functions.
We will see in the next section that the lower semicontinuity of the functionals presented in this sections work for domains with Lipschitz continuous boundary with the same conditions as for the boundary, but the tools used in proving the lower semicontinuity are completely different.

March 1, 2013 at 12:47 pmMaster 5  Beni Bogoşel's blog