Master 6
(For the context see the Shape Optimization page where you can find links to the first 5 parts)
A particular consequence of the ModicaMortola Theorem is that the functional
is lower semicontinuous with respect to the convergence for on the set
where the equalities are, as usual, up to a set of measure zero. It would be nice if a similar result would be true for multiphase systems, where a functional of the form
is a limit and therefore semicontinuous, for where
Let’s first remark that allowing the function in the ModicaMortola theorem to have more than two zeros does not suffice. Indeed, if we allow to have zeros , then the limiting phase will take only two values and or and , depending on the constraint . This means that functionals of the form we presented above cannot be represented as a limit when the function is scalar, but with more than two zeros. This obstacle can be overcome by passing to the multidimensional case. This approach is presented by Sisto Baldo in [1] and we will present the ideas of this approach below.
Consider , be an open bounded set with Lipschitz continuous boundary. Take such that for every with the constraint
where is given. We also take a function , with exactly zeros . () We assume that satisfies the condition
We make a technical assumption on : there exist such that
We will need another condition on which will allow us to correct the volume when constructing an approximating sequence in the proof of the (LS) property. For this one of the two conditions mentioned below will suffice:
 (a) is bounded from above.
 (b) converges superlinear to zero near all of the , i.e. for all there exists a small ball centered at and some numbers such that
Also define the following metric on :
It is easy to see that does not depend on the parametrization of the path ; this is a simple consequence of the change of variables property.
Remark 1 The definition of the metric above is equivalent to
where we have denoted by the space of paths which are Lipschitz continuous. The restriction of the definition to paths is good for the proof of the theorem from the article of Baldo [1], but the space of functions does not have sufficiently strong compactness properties we will need in the final of this section. A sequence of paths which converges uniformly, does not necessarily have a limit, but a sequence of Lipschitz continuous paths, for which the Lipschitz constants are bounded from above, if it converges uniformly to a path, then this path is also Lipschitz continuous.Note that a continuous path is ‘nice’ enough, in the sense that it has a well defined length if it is rectifiable, i.e. it has bounded variation. Restricting ourselves to the class of Lipschitz path, not only do we get a well defined length for our path, but we also know that the path is almost everywhere differentiable, and the fundamental formula of calculus works. Furthermore, Lipschitz continuous paths, as well as paths have an arclength parametrization, which allows us to parametrize them with constant speed on . For more details see [2] Chapter 4.
To see that the two definitions of the metric are equivalent, it is enough to pick a Lipschitz continuous path and show that
can be approximated as well as we want by where is a path. For this note that so we can approximate in the norm by a continuous function such that . Define by . From here we deduce right away that is and for every we have
Using the above estimates and the fact that the paths close to the infimum must be contained in a compact set as a consequence of 1, it follows that the two definitions are equivalent. In view of this fact, we will use in each case the definition which suits best our needs.
Using the above metric we can define for the following functions:
The following proposition states a couple of properties of the metric .
Proposition 1 The function is locally Lipschitz continuous. Moreover, if , then and the following inequality holds:
Proof: First, it is easy to see that because is a metric we have for every and for every
Now let’s estimate .
Suppose is a compact, and define to be the convex hull of which is also a compact and . Since
by picking, for example we get that
This proves that is locally Lipschitzcontinuous for all .
If the inequality
holds for every open set then the proposition is proved by using the monotone convergence theorem.
Consider the case . The function is locally Lipschitzcontinuous, and therefore differentiable almost everywhere in . Take a differentiability point of , be a sequence converging to and be the segment . By the definition of the metric we have
where in the last equality we have applied the mean value theorem and , i.e. . ( denotes the segment between and for every ) Divide the inequality obtained by and take the limit as to get
Take , and let be a sequence such that in . By passing, eventually, to subsequences we can assume that and almost everywhere in . Take in . Then we have
where we have used the dominated convergence theorem, the inequality obtained for functions and Fatou’s Lemma. This finishes the proof of the proposition. \hfill
Given two regular positive Borel measures and on , we define the supremum as the smallest regular positive measure which is greater or equal to and on all Borel subsets of . We have
In the same way we can define recursively the supremum of more than two measures defined on .
Consider a function such that almost everywhere and . This implies that
where are pairwise disjoint subsets of such that . The next proposition proves that the supremum of some well chosen measures on is the expression we are looking for.
Proposition 2 Denote the Borel measure . Then for every and
Proof: First, let’s prove that the sets have finite perimeter. We apply the coareatype FlemingRishel formula.
where . This implies that for . We present now a lemma which will help us prove the proposition.
Lemma 3 Let be a regular Borel measure on and be disjoint Borel subsets of with finite measure and be positive coefficients. Define
Then .
The proof of the lemma is quite simple. It suffices to notice that for every , and for each open set for some we have
Let’s now return to the proof of the proposition. Pick an open subset of . Then for every we have
and we also have
where . A simple inductive argument on the cardinality of proves that
Pick and suppose for simplicity that
By the FlemingRishel coarea formula and the above result we obtain
The last part of the above equalities was obtained by changing the order of summation. Notice that the coefficient of depends only on and because of the supposed ordering it is equal to . The same result can be obtained for every . Therefore for every we have the following equality
Note that for every , because of the definition of we have
for every with equality if and only if or . Now we can apply the above lemma for and
Because , from the lemma we find that
from which the desired result follows if we take . \hfill
Using the result of the previous proposition we define for every
and notice that when we have with , , and
Define, also, for every and for any
Now we are able to state the main result of this section.
Theorem 4 (Baldo) Under the above considerations we have
Proof: As in every convergence proof, we split the proof in two parts: the proof of (LI) and the proof of (LS). We first approach the proof of (LI), which is immediate, due to Propositions 1 and 2.
Pick and a sequence such that in . Take a sequence . It is not restrictive to assume that exists and is finite. We can choose a subsequence that converges to pointwise almost everywhere in , and by Fatou’s lemma we have
Since is continuous and nonnegative it follows that almost everywhere in . Now we need to prove that
Using the assumption (1) on we can further reduce the problem to the case is equibounded, i.e. replace each scalar component of with its truncation with respect to and . To simplify the notations, we denote the truncated sequence like the initial sequence. Since almost everywhere in we see that takes, up to a set of measure zero, only the values which all are in . This means that the truncation does not affect the convergence of to . Note also that by truncation the value of the integrals in the right hand side of (2) decrease, so we are going to prove a stronger inequality. Since is locally Lipschitzcontinuous it follows that in for every . By the lower semicontinuity of the total variation we have for every
for every open set . In the following formulas the supremums are taken over all families of pairwise disjoint open subsets of .
Using the result of Proposition 1 we get
Combining the above results we obtain
and the proof of (LI) is complete.
Let’s now turn to the proof of the (LS) property. Pick . If then any sequence which converges to in will satisfy (LS). Therefore, without loss of generality we can assume that
where are pairwise disjoint sets with finite perimeter in such that .
The following lemma, whose proof can be found in the article of Baldo [1], allows us to consider only partitions with the property that are polygonal domains in with .
Lemma 5 Let be a partition of like in the expression of . Then there exists a sequence of partitions of such that
 (i) is a polygonal domain and for any and for all ;
 (ii) If then in as ;
 (iii) for all ;
 (iv) .
We need another lemma which generalizes an idea of Modica [4]. We assume for simplicity that for every there exists a distance minimizing geodesic connecting and , i.e. we suppose that there exists a path such that and
We will see in the end of the proof how to modify the proof if such geodesics do not exist. Also note that since the value of is independent of the parametrization of the path , we can assume that for every .
Lemma 6 Consider the following family of differential equations:
for and fixed.Then, for every there exists a Lipschitz continuous function and three constants and depending only on such that:
 (i)
 (ii) almost everywhere in .
 (iii) If then on the set
depends only on and we can write
for any such that , where solves (3).
Proof of the lemma 6: We only need to find the constants and and to define at the points different from those considered in (i). Define
and note that is increasing. If we denote , we immediately obtain . The inverse function of satisfies the differential equation (3). We extend this function to the whole by putting
Therefore is a Lipschitz continuous function satisfying (3) in every point where . Putting
we can define as required on the strips considered in (iii). We choose
and
Then and on the subsets of described in (i) and (iii). Using an extension result for Lipschitz continuous functions (for example Kirszbraun’s theorem, see [3]) we can extend to the whole and the extension has the same Lipschitz constant. Moreover, Rademacher’s theorem says that a Lipschitz map is almost everywhere differentiable, and its differential (at the points where it exists) is bounded by the best Lipshitz constant for our function. Therefore we can choose . This ends the proof of the lemma.
We state here another Lemma following an idea of [4] which is used in our proof:
Lemma 7 Let a polygonal domain and an open subset of such that . Define by
Then is Lipschitz continuous, for almost all and if then
We will now prove the (LS) property for domains with properties given in Lemma 5. Note that for sets having the properties mentioned in the lemma we can apply the results of Lemma 7. We define for
We fix and note that for small enough we have almost everywhere on the set , for all . Consider the sequence of functions
We denote and using the coarea formula we obtain
where and because of Lemma 7 we have
This means that in . If then define . Otherwise denote
and find that , where is a constant.
We denote
We will now define such that , and equals in all of except a small set, such that the values of on that small set will not affect the behavior of as . We will give two variants for the definition of , using the hypotheses (a) or (b) on .
Remark 2 In the original article of Baldo [1] the hypotheses presented below are not present, and the proof presented in the article contains an error in this part where we build the recovery sequence and correct its integral. The same error is present in the article [4] of Modica. Hypothesis (a) is a quick fix, which is quite restrictive for (no polynomial functions are allowed) even though, bu truncating very far from its zeros by a constant function should not modify the convergence result.Hypothesis (b) was suggested by Sisto Baldo in a short email discussion I had with him about the absence of some necessary additional hypothesis for his proof to work.
Hypothesis (a) is bounded from above by .
In this case we pick an open ball such that is an interior point for (if this set is void, we relabel such that has the desired property). For small enough we have . Therefore we may define as follows:
where is chosen such that
We have
where is the volume of the dimensional unit ball. Therefore we define
and note that
We evaluate
and note that for the limit in the right hand side is zero. Therefore we may pick in the prescribed interval, which is nonvoid for , and we are done.
Hypothesis (b) converges superlinear to zero near all of the , i.e. for all there exists a small ball centered at and some numbers such that
We pick an open ball where is an interior point of , and is small enough and fixed such that . Note that for small enough we have . We define as follows:
where is chosen such that . It is easy to see that , where are positive constants. It is still true that in . It remains to calculate:
because .
This new function satisfies the constraint for small enough. To be able to give a good estimate for we consider the following partition of :
We obviously have
We now discuss each of the terms in the right hand side of the above inequality. First, it is easy to note from the definition of and that for small enough we have on , and this means that all the terms from the sum
are zero.
We turn now to the second term, and note that by either of the constructions (a) or (b) we have
We go now to the third term and define . Since , by using Lemma 6 we find that there exists a constant such that
Denote and using again the coarea formula we have
For almost all we have , and therefore, using Lemma 7 we obtain:
Passing to the infimum for we get that
and this implies
It remains now to estimate the terms of the form
Using the coarea formula and (3) we obtain
where
and by Lemma 7 we have
By passing to as in the above inequalities we get
Passing now to infimum as we get that
and by a diagonal argument the proof of (LS) property is finished.
The proof is now almost done. We assumed that geodesics between exist, but if such geodesics do not exist, then we can choose approximate geodesics such that
and reasoning as above we can construct a sequence such that
and again a diagonal argument finishes the proof. \hfill
[1] Sisto Baldo, Minimal Interface Criterion for Phase Transitions in Mixtures of CahnHilliard fluids [2] Leoni, Giovanni, A {F}irst {C}ourse in {S}obolev {S}paces, {Graduate {S}tudies in {M}athematics, {V}ol 105}, {American Mathematical Society}
[3] Federer, Herbert, Geometric measure theory
[4] Modica, Luciano, Gradient Theory of Phase Transitions with Boundary Contact Energy

March 3, 2013 at 1:52 pmMaster 7  Beni Bogoşel's blog