Home > shape optimization > Modica-Mortola Theorem

Modica-Mortola Theorem

March 31, 2012 Leave a comment Go to comments

The notion of {\Gamma}-convergence was introduced by E. De Giorgi and T. Franzioni in the article Su un tipo di convergenza variazionale 1975.

Let {X} be a metric space, and for {\varepsilon >0} let be given {F_\varepsilon : X \rightarrow [0,\infty]}. We say that {F_\varepsilon} {\Gamma}-converges to {F} on {X} as {\varepsilon \rightarrow 0}, and we write {\Gamma-\lim F_\varepsilon =F}, if the following conditions hold:

  • (LI) For every {u \in X} and every sequence {(u_\varepsilon)} such that {u_\varepsilon \rightarrow u} in {X} we have

    \displaystyle \liminf_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\geq F(u)

  • (LS) For every {u \in X} there exists a sequence {(u_\varepsilon)} such that {u_\varepsilon \rightarrow u} in {X} and

    \displaystyle \limsup_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\leq F(u).

In practice, the (LI) property is often easy to verify, but the property (LS) rises more problems, because the sequence {u_\varepsilon} must be constructed for every {u}. It is often convenient to find a set {\mathcal D \subset X}, which satisfies the following conditions for every {u \in X} there is an approximating sequence {(u_n) \subset \mathcal D} such that {u_n \rightarrow u} and {F(u_n) \rightarrow F(u)}; then a simple diagonal argument shows that is enough to verify (LS) only for elements {u \in \mathcal{D}}. We can push this argument a bit further, and just verify that for every {u \in \mathcal D} and every {\eta>0} there is a sequence {(u_\varepsilon) \subset X} such that {\limsup d(\varepsilon, u)\leq \eta} and {\limsup F_\varepsilon(u_\varepsilon) \leq F(u)+\eta}.

The {\Gamma}-convergence has the following properties:

  • (i) The {\Gamma}-limit {F} is always lower semicontinuous on {X};
  • (ii) {\Gamma}-convergence is stable under continuous perturbations: if {F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F} and {G} is continuous, then

    \displaystyle F_\varepsilon + G \stackrel{\Gamma}{\longrightarrow} F+G

  • (iii) If {F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F} and {v_\varepsilon} minimizes {F_\varepsilon} over {X}, then every limit point of {(v_\varepsilon)} minimizes {F} over {X}.

Consider a container which is filled with two immiscible and incompressible fluids. In the classical theory of phase transition is assumed that, at equilibrium, the two fluids arrange themselves in order to minimize the area of their interface wich separates the two phases (here we neglect the gravity and the interaction of the fluids with the wall of the contained). This situation is modelled as follows: the contained is represented by a bounded regular domain {\Omega \subset \Bbb{R}^N} ({N=3} is the physical case), and every configuration of the system is described by a function {u} on {\Omega} which takes the value {0} on the set occupied by the first fluid and the value {1} on the set occupied by the second fluid. The set of discontinuities of {u} is the interface between the two fluids, and we denote it by {Su}. The set of admissible configurations is given by all {u :\Omega \rightarrow \{0,1\}} which satisfy {\int_\Omega u=c} where {c} is the volume of the second fluid, {c \in (0, |\Omega|)}. Ignoring the gravity and the contact with the walls, we can assume that the energy of the system has the form {F(u)=\sigma \mathcal{H}^{N-1}(Su)} where {\sigma} is the interfacial tension between the two fluids and {\mathcal{H}^{N-1}} is the {N-1} dimensional Hausdorff measure. Therefore {F(u)} is a surface energy distributed on the interface {Su}, and the equilibrium configuration is obtained by minimizing {F} over the space of admissible configurations.

A different way of studying systems of two immiscible fluids is to assume that the transition is not given by a separating interface, but is rather a continuous phenomenon occuring in a thin layer which we identify with the interface. This means that we allow a fine mixture of the two fluids. In this case a configuration of the system is represented by a function {u :\Omega \rightarrow [0,1]} where {u(x)} represents the average volume density of the second fluid at the point {x \in \Omega}. ({u(x)=0} means that only the first fluid is present at {x}, {u(x)=1/2} means that both fluids are present near {x} with the same rate, {u(x)=1} means that only the second fluid is present at {x}, etc.) The space of admissible configurations is the class {X} of all {u : \Omega \rightarrow [0,1]} such that {\int_\Omega u=c}, and to every configuration {u} we associate the energy

\displaystyle E_\varepsilon(u)=\int_\Omega \left[\varepsilon^2 |Du|^2+W(u) \right]dx

where {\varepsilon} is a small parameter and {W} is a continuous positive function which vanishes only at {0} and {1} (a double-well potential). If we want to minimize {E_\varepsilon} the term {\int W(u)} favors configurations which take only values close to {0} and {1}, while the term {\varepsilon^2 \int_\Omega |Du|^2} penalizes the size of the transition layer between the states {0} and {1}. When {\varepsilon} is small the first term is also small, and the minimum of {E_\varepsilon} will, roughly, take only the values {0} and {1} (it must take both because of the constraint {\int_\Omega u=c}) and the transition only occurs in a thin layer of order {\varepsilon}. This model was proposed by J.W. Cahn and J.E. Hilliard in Free Energy of a Nonuniform System. I. Interfacial Free Energy, from Journal of Chemical Physics, 1958.

A connection between the classical model and the Cahn-Hilliard model was established by L. Modica (in the article Gradient Theory of Phase Transitions with Boundary Contact Energy, 1987 ), who proved that minimizers of {E_\varepsilon} converge to minimizers of {F}. This was obtained by proving that suitable rescalings of the functionals {E_\varepsilon} {\Gamma}-converge to {F}. Note that for the mathematical model it is not essential that the phases of the liquid are {0} and {1}. We may just as well take any two real values {\alpha<\beta} for the phases, consider {\phi :\Bbb{R} \rightarrow \Bbb{R}, \phi(x)=\alpha+(\beta-\alpha)x}, and define

\displaystyle u'= \phi\circ u,\ W'=W\circ \phi^{-1}.

The only modification in the formula of {E_\varepsilon} is a constant factor which comes from {Du'} and which can be imbedded in {\varepsilon}. Therefore, without loss of generality we can assume any real values we want for the phases {\alpha, \beta}, and the result remains the same. In the variant of the theorem presented here we will take {W: \Bbb{R} \rightarrow [0,\infty)} with only two zeros at {-1} and {1}. Also, define

\displaystyle C_0=2 \int_{-1}^1 \sqrt{W(s)}ds.

Consider {\Omega \subset \Bbb{R}^N} a bounded open set with Lipschitz boundary. Define for every {u \in L^1(\Omega)}

\displaystyle F_\varepsilon(u)=\frac{1}{\varepsilon}E_\varepsilon(u)=\begin{cases} \int_\Omega \left[\varepsilon |Du|^2 +\frac{1}{\varepsilon}W(u)\right]dx & u \in H^1(\Omega),\ \int_\Omega u=c \\ \infty & \text{otherwise}\end{cases}

and

\displaystyle F(u)=\begin{cases} C_0 \text{Per}_\Omega(E_1) & u \in BV(\Omega; \{-1,1\}), \ E=u^{-1}(1),\ \int_\Omega u=c \\ \infty & \text{otherwise} \end{cases}

I have adapted the proof below using ideas from Modica’s article and from the lecture notes from a course on \Gamma-convergence theory given by Giuseppe Buttazzo. The notes can be found here.

Modica Mortola Theorem We have {F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F} in {L^1(\Omega)}.

Proof: It is convenient to introduce the function

\displaystyle \phi(t)=\int_0^t \sqrt{W(s)}ds

and note that {C_0=2(\phi(1)-\phi(-1))}. It is easy to obtain from the definition of the total variation that if

\displaystyle v \in BV(\Omega;\{a,b\}),\ v(x)=\begin{cases} a & x \in E \\ b & x \in \Omega\setminus E\end{cases}

then {\text{Per}_\Omega(E)<\infty} and

\displaystyle \int_\Omega |Dv|=|\alpha-\beta|\text{Per}_\Omega(E).

Therefore, for {u \in BV(\Omega; \{-1,1\})} we can write

\displaystyle 2 \int_\Omega |D(\phi\circ u)|=2(\phi(1)-\phi(-1))\text{Per}_\Omega(u^{-1}(1))=F(u).

We are now ready to prove the (LI) property from the {\Gamma}-convergence definition. Take {u \in L^1(\Omega)} and {(u_n) \subset L^1(\Omega),\ u_\varepsilon \rightarrow u} in {L^1(\Omega)}. It is not restrictive to assume that {F_\varepsilon(u_\varepsilon)} and {\liminf_{\varepsilon \rightarrow 0} F_\varepsilon(u_\varepsilon)} are finite, because otherway the inequality is trivial to prove. The {L^1(\Omega)} convergence of {(u_\varepsilon)} to {u} implies the existence of a subsequence {(u_{\varepsilon_n})} which converges almost everywhere to {u} as {\varepsilon_n \rightarrow 0} on {\Omega}. By Fatou’s Lemma and the continuity of {W} we have

\displaystyle \int_\Omega W(u(x))dx \leq \liminf_{\varepsilon_n \rightarrow 0}\int_\Omega W(u_{\varepsilon_n}(x))dx\leq \liminf_{\varepsilon_n \rightarrow 0}\varepsilon_n F_{\varepsilon_n}(u_{\varepsilon_n})=0,

which implies {W(u(x))=0} almost everywhere on {\Omega}, and therefore {|u| =1} almost everywhere on {\Omega}. This leads us to

\displaystyle \liminf_{\varepsilon \rightarrow 0} F_\varepsilon(u_\varepsilon) \geq \liminf_{\varepsilon \rightarrow 0} \int_\Omega 2|Du_\varepsilon|\sqrt{W(u_\varepsilon)}dx=

\displaystyle =\liminf_{\varepsilon \rightarrow 0} \int_\Omega 2|D(\phi\circ u_\varepsilon)|\geq 2 \int_\Omega|D(\phi\circ u)|

where we have used the inequality {a^2+b^2 \geq 2ab} and the lower semicontinuity of the total variation. This proves the (LI) property.

We now turn to the (LS) property. The case when {F(u)=\infty} is trivial, therefore we may assume that {F(u)<\infty}, and we want to find a dense subset {\mathcal{D} \subset BV(\Omega;\{-1,1\})} such that for every {u \in BV(\Omega;\{-1,1\})} there is an approximating sequence {u \in \mathcal{D}} such that {F(u_n) \rightarrow F(u)}. One such subset may be found by looking at Lemma 1 from the article of Luciano Modica, The gradient theory of phase transitions and the minimal interface criterion, 1987 which is stated bellow.

Lemma 1.

Let {\Omega} be an open, bounded subset of {\Bbb{R}^N} with Lipschitz continuous boundary, and let {E} be a measurable subset of {\Omega}. If {E} and {\Omega\setminus E} both contain a non-empty open ball, then there exists a sequence {(E_h)} of open bounded sets of {\Bbb{R}^N} with smooth boundaries such that

  • (i) {\displaystyle \lim_{h \rightarrow \infty}|(E_h \cap \Omega)\Delta E|=0,\ \lim_{h \rightarrow \infty} \text{Per}_\Omega(E_h)=\text{Per}_\Omega(E) };
  • (ii) {|E_h \cap \Omega|=|E|} for {h} large enough;
  • (iii) {\mathcal{H}^{N-1}(\partial E_h \cap \partial \Omega)=0} for {h} large enough.

In our case {u} has the form {-\chi_A+\chi_{\Omega \setminus A}}, where {A} is a subset of {\Omega} with bounded perimeter. Using the above lemma we could define

\displaystyle \mathcal{D}=\{-\chi_A+\chi_{\Omega \setminus A} : A \subset \Omega, A \text{ open with smooth boundary}, \mathcal{H}^{N-1}(\partial A \cap \partial \Omega)=0\},

if we knew that every bounded perimeter set {E \subset \Omega} has the property that {E} and {\Omega \setminus E} both contain a non-empty open ball. This is not true in general, but Theorem 1.24 from Enrico Giusti. Minimal surfaces and functions of bounded variation says that any finite perimeter set {E} can be approximated by sets with smooth boundary {(E_h)} such that

\displaystyle \lim_{h \rightarrow \infty} \int_\Omega|\chi_{E_h}-\chi_E|=0 \text{ and } \lim_{h \rightarrow \infty} \text{Per}_\Omega(E_h)=\text{Per}_\Omega(E).

This reduces the problem to proving (LS) for all {u \in \mathcal{D}}.

In the following we need another lemma from the paper L. Modica, The gradient theory of phase transitions and the minimal interface criterion.

Lemma 2.

Let {A} be an open set of {\Bbb{R}^N} with smooth, non-empty, compact boundary and {\Omega} an open subset of {\Bbb{R}^N} such that {\mathcal{H}^{N-1}(\partial A \cap \partial \Omega)=0}. Define {h : \Bbb{R}^N \rightarrow \Bbb{R}} by

\displaystyle h(x)=\begin{cases} -d(x,\partial A) & x \in A \\ \hfill d(x,\partial A) & x \notin A \end{cases}.

Then {h} is Lipschitz continuous, {|Dh(x)|=1} for almost all {x \in \Bbb{R}^N} and if {S_t=\{x \in \Bbb{R}^N : h(x)=t\}} then

\displaystyle \lim_{t \rightarrow 0}\mathcal{H}^{N-1}(S_t \cap \Omega)=\mathcal{H}^{N-1}(\partial A \cap \Omega).

Take {u=-\chi_{A \cap \Omega}+\chi_{\Omega \setminus A}} such that {\int_\Omega u=c}, where {A} is an open set with smooth boundary such that {\mathcal{H}^{N-1}(\partial A \cap \partial \Omega)=0}. We do assume that {A\cap \Omega \neq \emptyset}.

Set for every {t \in \Bbb{R}}

\displaystyle \psi_\varepsilon(t)=\int_{-1}^t \frac{\varepsilon}{\sqrt{\varepsilon+W(s)}}ds

\displaystyle \varphi_\varepsilon(t)=\begin{cases}-1 & t \leq 0 \\ \psi_\varepsilon^{-1}(t) & 0 \leq t \leq \psi_\varepsilon(1)\\ 1 & t \geq \psi_\varepsilon(1) \end{cases}

and define

\displaystyle u_\varepsilon(x)=\varphi_\varepsilon(d(x)+\eta_\varepsilon)

where

\displaystyle d(x)=\begin{cases}-d(x,\partial A) & x \in A \\\hfill d(x,\partial A) & x \notin A \end{cases}

and \eta_\varepsilon is chosen such that \int_\Omega u_\varepsilon(x)dx=c.

Note that for {d(x)\leq 0} or {d(x) \geq \psi_\varepsilon(1)} we have {u_\varepsilon(x)=u(x)} and therefore, using the coarea formula we get

\displaystyle \int_\Omega |{u}_\varepsilon(x) -u(x)|dx=\int_{0\leq d(x)\leq \psi_\varepsilon(1)} |\varphi_\varepsilon(d(x))-1||\nabla d(x)|dx=

\displaystyle =\int_0^{\psi_\varepsilon(1)}|\varphi_\varepsilon(t)-1|\mathcal{H}^{N-1}(\{x \in \Omega: d(x)=t\})dt \leq\\ \leq 2\psi_\varepsilon(1)\sigma_{\psi_\varepsilon(1)}\leq4\sqrt{\varepsilon}\sigma_{2\sqrt{\varepsilon}}

where

\displaystyle \sigma_a=\sup_{-a\leq t \leq a} \mathcal{H}^{N-1}(\{x \in \Omega: d(x)=a\}).

Notice that by Lemma 2 we have {\sigma_a \rightarrow \mathcal{H}^{N-1}(\partial A \cap \Omega)} as {a \rightarrow 0} and therefore {{u}_\varepsilon \rightarrow u} in {L^1(\Omega)}.

Denote {\Sigma_\varepsilon=\{x \in \Omega : -\eta_\varepsilon <d(x)<\psi_\varepsilon(1)-\eta_\varepsilon\}} and notice that if {\varepsilon} is small enough then {|\nabla d(x)|=1} for all {x \in \Sigma_\varepsilon}. Using the coarea formula we get

\displaystyle F_\varepsilon(u_\varepsilon)=\int_\Omega \left[\varepsilon |\varphi_\varepsilon ' (d(x)+\eta_\varepsilon)|^2+\frac{1}{\varepsilon}W(\varphi_\varepsilon(d(x)+\eta_\varepsilon)) \right]dx =

\displaystyle = \int_{\Sigma_\varepsilon} \left[ \varepsilon |\varphi_\varepsilon ' (d(x)+\eta_\varepsilon)|^2+\frac{1}{\varepsilon}W(\varphi_\varepsilon(d(x)+\eta_\varepsilon)) \right]dx=

\displaystyle = \int_{-\eta_\varepsilon}^{\psi_\varepsilon(1)-\eta_\varepsilon} \left[\varepsilon |\varphi_\varepsilon ' (t+\eta_\varepsilon)|^2+\frac{1}{\varepsilon}W(\varphi_\varepsilon(t+\eta_\varepsilon) )\mathcal{H}^{N-1}(\{x \in \Omega : d(x)=t\})\right] dt \leq

\displaystyle \leq \sigma_{\psi_\varepsilon(1)} \int_0^{\psi_\varepsilon(1)} \left[\varepsilon |\varphi_\varepsilon ' (t)|^2+\frac{1}{\varepsilon}W(\varphi_\varepsilon(t) ) \right]dt

By the definitions of {\psi_\varepsilon, \varphi_\varepsilon} we have

\displaystyle \varphi_\varepsilon'=\frac{1}{\psi_\varepsilon'(\psi_\varepsilon^{-1})}=\frac{\sqrt{\varepsilon+W(\psi_\varepsilon^{-1})}}{\varepsilon}=\frac{1}{\varepsilon}\sqrt{\varepsilon+W(\varphi_\varepsilon)}.

Using this and the previous relations we obtain

\displaystyle F_\varepsilon(u_\varepsilon)\leq \sigma_{\psi_\varepsilon(1)} \int_0^{\psi_\varepsilon(1)} \left[\frac{\varepsilon+W(\varphi_\varepsilon)}{\varepsilon}+\frac{1}{\varepsilon}W(\varphi_\varepsilon(t) ) \right]dt\leq

\displaystyle \leq \frac{2\sigma_{\psi_\varepsilon(1)}}{\varepsilon} \int_0^{\psi_\varepsilon(1)}[\varepsilon+W(\varphi_\varepsilon)]dt=

\displaystyle =2\sigma_{\psi_\varepsilon(1)}\int_{-1}^1 \sqrt{\varepsilon+W(s)}ds

where in the last equality we have changed the variable {s=\varphi_\varepsilon(t)}. Taking now {\limsup} as {\varepsilon \rightarrow 0} in the inequality obtained and using Lemma 2 we get

\displaystyle \limsup_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\leq C_0 \text{Per}_\Omega(A)=F(u).

This proves (LS) and finishes the proof. {\square}

Categories: shape optimization Tags: , , ,