Geometric Minimizing Movements

Abstract

Geometric motions; i.e., motions of sets governed by geometric quantities, such as mean curvature flow, can sometime be cast in the framework of minimizing movements following the approach of Almgren, Taylor and Wang. In this case the energy is a perimeter functional, and the distance term must be suitably rewritten as an anisotropic integral. With this approach, it is possible to prove existence for motions by curvature; e.g., mean curvature or crystalline curvature. For oscillating perimeter energies, we apply the approach of the minimizing movements along a sequence, computing an effective motion showing pinning for large sets and a discontinuous dependence of the velocity on the curvature. Additional effects are shown to appear for minimizing movements related to the homogenization of perimeter functionals with oscillating forcing terms.

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We now examine some minimizing movements describing the motion of sets. Such a motion can be framed in the setting of the previous chapter after identification of a set A with its characteristic function u =χ _A . The energies we are going to consider are of perimeter type; i.e., with

$$\displaystyle{ F(A) = {\mathcal{H}}^{n-1}(\partial A) }$$

(9.1)

as a prototype in the notation of the previous chapter.

9.1 Motion by Mean Curvature

The prototype of a geometric motion is motion by mean curvature; i.e., a family of sets A(t) whose boundary moves in the normal direction with velocity proportional to its curvature (inwards in convex regions and outwards in concave regions). In the simplest case when the initial datum is a ball in ${\mathbb{R}}^{2}$, $A(0) = A_{0} = B_{R_{0}}(0)$, the motion is given by concentric balls with radii satisfying

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} R^{\prime} = -\frac{c} {R} \quad \\ R(0) = R_{0};\quad \end{array} \right. }$$

(9.2)

i.e., $R(t) = \sqrt{R_{0 }^{2 } - 2ct}$, valid until the extinction time $t = R_{0}^{2}/2c$, when the radius vanishes.

A heuristic argument suggests that the variation of the perimeter be linked to the notion of curvature; hence, we expect to be able to obtain motion by mean curvature as a minimizing movement for the perimeter functional. We will see that, in order to obtain geometric motions as minimizing movements, we will have to modify the procedure described in the previous chapter.

Example 9.1 (Pinning for the perimeter motion).

Let n = 2. We apply the minimizing-movement procedure to the perimeter functional (9.1) and the initial datum $A_{0} = B_{R_{0}}(0)$ in ${\mathbb{R}}^{2}$.

With fixed τ, since

$$\displaystyle{\int _{{\mathbb{R}}^{2}}\vert \chi _{A} -\chi _{B}{\vert }^{2}\,\mathit{dx} = \vert A\bigtriangleup B\vert,}$$

the minimization to determine A ₁ is

$$\displaystyle{ \min {\Bigl \{{\mathcal{H}}^{1}(\partial A) + \frac{1} {2\tau }\vert A\bigtriangleup A_{0}\vert \Bigr \}}. }$$

(9.3)

We note that we can restrict our attention to sets A contained in A ₀, since otherwise taking $A \cap A_{0}$ as test sets in their place would decrease both terms in the minimization. Once this is observed, we also note that, given A ⊂ A ₀, if B _R (x) ⊂ A ₀ has the same measure as A then it decreases the perimeter part of the energy (strictly, if A itself is not a ball) while keeping the second term fixed. Hence, we can limit our analysis to balls B _R (x) ⊂ A ₀, for which the energy depends only on R. The incremental problem is then given by

$$\displaystyle{ \min {\Bigl \{2\pi R + \frac{\pi } {2\tau }(R_{0}^{2} - {R}^{2}): 0 \leq R \leq R_{ 0}\Bigr \}}, }$$

(9.4)

whose minimizer is either R = 0 (with value $\frac{\pi }{2\tau }R_{0}^{2}$) or R = R ₀ (with value 2π R ₀), since in (9.4) we are minimizing a concave function of R. For τ small the minimizer is then R ₀. This means that the motion is trivial: A _k = A ₀ for all k, and hence also the resulting minimizing movement is trivial.

9.2 A First (Unsuccessful) Generalization

We may generalize the scheme of the minimizing movements by taking a more general distance term in the minimization; e.g., considering x _k as a minimizer of

$$\displaystyle{ \min {\Bigl \{F(x) + \frac{1} {\tau } \varPhi (\|x - x_{k-1}\|)\Bigr \}}, }$$

(9.5)

where Φ is a continuous increasing function with Φ(0) = 0. As an example, we can consider

$$\displaystyle{\varPhi (z) = \frac{1} {p}\vert z{\vert }^{p}.}$$

Note that in this case we obtain the estimate

$$\displaystyle{\|x_{k} - x{_{k-1}\|}^{p} \leq p\,\tau (F(x_{ k-1}) - F(x_{k}))}$$

for the minimizer x _k . Using Hölder’s inequality as in the case p = 2, we end up with (for j > h)

$$\displaystyle\begin{array}{rcl} \|x_{j} - x_{h}\|& \leq & {(j - h)}^{(p-1)/p}{\Bigl (\sum _{ k=h+1}^{j}\|x_{ k} - x{_{k-1}\|{}^{p}\Bigr )}}^{1/p} {}\\ & \leq & {(p\,F(x_{0}))}^{1/p}{{(\tau }^{1/(p-1)}(j - h))}^{(p-1)/p}. {}\\ \end{array}$$

In order to obtain the ($1 -\frac{1} {p}$)-Hölder continuity for the interpolated function u ^τ, we have to define it as

$$\displaystyle{{u}^{\tau }(t) = u_{\lfloor t{/\tau }^{1/(p-1)}\rfloor }.}$$

Note that we may use the previous definition ${u}^{\tau }(t) = u_{\lfloor t/\tau \rfloor }$ with time step τ for the interpolated function if we change the parameter τ in (9.5) and, to define x _k , consider the problem

$$\displaystyle{ \min {\Bigl \{F(x) +{ \frac{1} {\tau }^{p-1}}\varPhi (\|x - x_{k-1}\|)\Bigr \}} }$$

(9.6)

instead.

Example 9.2 ((non-)geometric minimizing movements).

We use the scheme above, with a slight variation in the exponents, since we will be interested in the description of the motion in terms of the radius of a ball in ${\mathbb{R}}^{2}$ (which is the square root of the L ²-norm and not the norm itself). As in the previous example, we take the initial datum $A_{0} = B_{R_{0}} = B_{R_{0}}(0)$, and consider A _k defined recursively as a minimizer of

$$\displaystyle{ \min {\Bigl \{{\mathcal{H}}^{1}(\partial A) + \frac{1} {{p\tau }^{p-1}}\vert A\bigtriangleup A_{0}{\vert }^{p}\Bigr \}}, }$$

(9.7)

with p > 1. As above, at each step the minimizer is given by balls

$$\displaystyle{ B_{R_{k}}(x_{k}) \subset B_{R_{k-1}}(x_{k-1}). }$$

(9.8)

The value of R _k is determined by solving

$$\displaystyle{ \min \Bigl \{2\pi R + \frac{{\pi }^{p}} {{p\tau }^{p-1}}{(R_{k-1}^{2} - {R}^{2})}^{p}: 0 \leq R \leq R_{ k-1}\Bigr \}, }$$

(9.9)

which gives

$$\displaystyle{ \frac{R_{k} - R_{k-1}} {\tau } = - \frac{1} {\pi R_{k}^{1/(p-1)}(R_{k} + R_{k-1})}. }$$

(9.10)

Note that, in this case, the minimum value is not taken at $R_{k} = R_{k-1}$ (this can be verified, e.g., by checking that the derivative of the function to be minimized in (9.9) is positive at R _k−1). By passing to the limit in (9.10) we deduce the equation

$$\displaystyle{ R^{\prime} = - \frac{1} {2\pi {R}^{p/(p-1)}}\, }$$

(9.11)

(valid until the extinction time).

Despite having obtained an equation for R, we notice that this approach is not satisfactory, since we have:

• (non-geometric motion) in (9.8) we have infinitely many solutions; namely, all balls centered in x _k with

  $$\displaystyle{\vert x_{k-1} - x_{k}\vert \leq R_{k-1} - R_{k}.}$$

  This implies that we may have moving centres x(t) provided that |x′|≤ R′ and x(0) = 0; in particular, we may choose $x(t) = (R_{0} - R(t))z$ for any z ∈ B ₁(0) which converges to R ₀ z; i.e., the point where the sets concentrate at the vanishing time may be any point in $\overline{B}_{R_{0}}$ at the extinction time. This implies that the motion is not a geometric one: sets do not move according to geometric quantities.

• (failure to obtain mean-curvature motion) even if we obtain an equation for R we never obtain the mean curvature flow since $p/(p - 1) > 1$.

9.3 A Variational Approach to Curvature-Driven Motion

In order to obtain motion by curvature, Almgren, Taylor and Wang have introduced a variation of the implicit-time scheme described above, where the term $\vert A\,\bigtriangleup A_{k}\vert$ is substituted by an integral term which favours variations which are ‘uniformly distant’ to the boundary of A _k . The problem defining A _k is then

$$\displaystyle{ \min {\Bigl \{{\mathcal{H}}^{1}(\partial A) + \frac{1} {\tau } \int _{A\bigtriangleup A_{k-1}}\text{dist}(x,\partial A_{k-1})\,\mathit{dx}\Bigr \}}. }$$

(9.12)

Note that the integral term can be indeed interpreted as an L ² distance between the boundaries of the sets.

We will not prove a general convergence result for an arbitrary initial datum A ₀, but we will check the convergence to mean-curvature motion for $A = B_{R_{0}}$ in ${\mathbb{R}}^{2}$.

In this case we note that if A _k−1 is a ball centered in 0 then we have:

• A _k is contained in A _k−1. To check this, note that, given a test set A, considering $A \cap A_{k-1}$ as a test set in its place decreases the energy in (9.12), strictly if $A\setminus A_{k-1}\neq \varnothing$.

• A _k is convex and with baricenter in 0. To check this, first, note that each connected component of A _k is convex. Otherwise, considering the convex envelopes decreases the energy (strictly, if one of the connected components if not convex). Then note that if 0 is not the baricenter of a connected component of A _k then a small translation towards 0 strictly decreases the energy (this follows by computing the derivative of the volume term along the translation). In particular, we only have one (convex) connected component.

From these properties we can conclude that A _k is indeed a ball centered in 0. Were it not so, there would be a line through 0 such that the boundary of A _k does not intersect perpendicularly this line. By a reflection argument, we then obtain a non-convex set $\tilde{A}_{k}$ with total energy not greater than the one of A _k (note that the line considered subdivides A _k into two subsets with equal total energy). Its convexification would then strictly decrease the energy. This shows that each A _k is of the form

$$\displaystyle{A_{k} = B_{R_{k}} = B_{R_{k}}(0).}$$

We can now compute the equation satisfied by R _k , by minimizing (after passing to polar coordinates)

$$\displaystyle{ \min {\Bigl \{2\pi R + \frac{2\pi } {\tau } \int _{R}^{R_{k-1} }(R_{k-1}-\rho )\rho \,d\rho \Bigr \}}, }$$

(9.13)

which gives

$$\displaystyle{ \frac{R_{k} - R_{k-1}} {\tau } = - \frac{1} {R_{k}}. }$$

(9.14)

Passing to the limit gives the desired mean curvature equation (9.2).

9.4 Homogenization of Flat Flows

We now consider geometric functionals with many local minimizers (introduced in Example 4.3) which give a more refined example of homogenization of minimizing movements. The functionals we consider are defined on (sufficiently regular) subsets of ${\mathbb{R}}^{2}$ by

$$\displaystyle{ F_{\varepsilon }(A) =\int _{\partial A}a{\Bigl (\frac{x} {\varepsilon } \Bigr )}d{\mathcal{H}}^{1}, }$$

(9.15)

where

$$\displaystyle{a(x_{1},x_{2}) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad &\text{if}\,x_{1} \in \mathbb{Z}\,\text{or}\,x_{2} \in \mathbb{Z}\\ 2\quad &\text{otherwise}. \end{array} \right.}$$

The Γ-limit of the energies $F_{\varepsilon }$ is the crystalline perimeter energy

$$\displaystyle{ F(A) =\int _{\partial A}\|\nu \|_{1}d{\mathcal{H}}^{1}, }$$

(9.16)

with $\|(\nu _{1},\nu _{2})\|_{1} = \vert \nu _{1}\vert + \vert \nu _{2}\vert$. A minimizing movement for F is called a flat flow. We will first briefly describe it, and then compare it with the minimizing movements for $F_{\varepsilon }$.

9.4.1 Motion by Crystalline Curvature

The incremental problems for the minimizing-movement scheme for F in (9.16) are of the form

$$\displaystyle{ \min {\Bigl \{F(A) + \frac{1} {\tau } \int _{A\bigtriangleup A_{k-1}}\text{dist}_{\infty }(x,\partial A_{k-1})\,\mathit{dx}\Bigr \}}, }$$

(9.17)

where for technical reasons we consider the ∞-distance

$$\displaystyle{\text{dist}_{\infty }(x,B) =\inf \{\| x - y\|_{\infty }: y \in B\}.}$$

However, in the simplified situation below this will not be relevant in our computations.

We only consider the case when the initial datum A ₀ is a rectangle, which plays the role played by a ball for motion by mean curvature. Note that, as in Sect. 9.3, we can prove that if A _k−1 is a rectangle, then we can limit the computation in (9.17) to

• A contained in A _k−1 (otherwise $A \cap A_{k-1}$ strictly decreases the energy).

• A with each connected component a rectangle (otherwise taking the least rectangle containing a given component would decrease the energy, strictly if A is not a rectangle).

• A connected and with the same center as A ₀ (since translating the center towards 0 decreases the energy).

Fig. 9.1

Incremental crystalline minimization

Hence, we may suppose that

$$\displaystyle{A_{k} ={\Bigl [ -\frac{L_{k,1}} {2}, \frac{L_{k,1}} {2} \Bigr ]} \times {\Bigl [-\frac{L_{k,2}} {2}, \frac{L_{k,2}} {2} \Bigr ]}}$$

for all k. In order to iteratively determine L _k , we have to minimize the energy

$$\displaystyle{ \min {\Bigl \{2(L_{k,1} +\varDelta L_{1}) + 2(L_{k,2} +\varDelta L_{2}) + \frac{1} {\tau } \int _{A\bigtriangleup A_{k-1}}\text{dist}_{\infty }(x,\partial A_{k-1})\,\mathit{dx}\Bigr \}}.\quad \, }$$

(9.18)

In this computation it is easily seen that for τ small the integral term can be substituted by

$$\displaystyle{\frac{L_{k,1}} {4} {(\varDelta L_{2})}^{2} + \frac{L_{k,2}} {4} {(\varDelta L_{1})}^{2}.}$$

This argument amounts to noticing that the contribution of the small rectangles at the corners highlighted in Fig. 9.1 is negligible as τ → 0. The optimal increments (more precisely, decrements) Δ L _j are then determined by the conditions

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} 1 + \frac{L_{k,2}} {4\tau } \varDelta L_{1} = 0 \quad \\ 1 + \frac{L_{k,1}} {4\tau } \varDelta L_{2} = 0.\quad \end{array} \right. }$$

(9.19)

Hence, we have the difference equations

$$\displaystyle{ \frac{\varDelta L_{1}} {\tau } = - \frac{4} {L_{k,2}},\qquad \frac{\varDelta L_{2}} {\tau } = - \frac{4} {L_{k,1}}, }$$

(9.20)

which finally gives the system of ODEs for the limit rectangles, with edges of length L ₁(t) and L ₂(t) respectively,

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} L_{1}^{\prime} = - \frac{4} {L_{2}} \quad \\ L_{2}^{\prime} = - \frac{4} {L_{1}}.\quad \end{array} \right. }$$

(9.21)

Geometrically, each edge of the rectangle moves inwards with velocity inversely proportional to its length; more precisely, equal to twice the inverse of its length (so that the other edge contracts with twice this velocity). Hence, in this context the inverse of the length of an edge plays the role of the curvature (crystalline curvature).

It is worth noticing that by (9.21) all rectangles are homothetic, since $\frac{d} {\mathit{dt}} \frac{L_{1}} {L_{2}} = 0$, and with area satisfying

$$\displaystyle{ \frac{d} {\mathit{dt}}L_{1}L_{2} = -8,}$$

so that $L_{1}(t)L_{2}(t) = L_{0,1}L_{0,2} - 8t$, which gives the extinction time $t = L_{0,1}L_{0,2}/8$. In the case of an initial datum a square of side length L ₀, the sets are squares whose side length at time t is given by $L(t) = \sqrt{L_{0 }^{2 } - 8t}$, in analogy with the evolution of balls by mean curvature flow.

9.5 Homogenization of Oscillating Perimeters

We consider the sequence $F_{\varepsilon }$ in (9.15). Note that, for any (sufficiently regular) initial datum A ₀, we have that $A^{\prime}_{\varepsilon } \subset A_{0} \subset A^{\prime\prime}_{\varepsilon }$, where $A^{\prime}_{\varepsilon }$ and $A^{\prime\prime}_{\varepsilon }$ are such that $F_{\varepsilon }(A^{\prime}_{\varepsilon }) = {\mathcal{H}}^{1}(\partial A^{\prime}_{\varepsilon })$ and $F_{\varepsilon }(A^{\prime\prime}_{\varepsilon }) = {\mathcal{H}}^{1}(\partial A^{\prime\prime}_{\varepsilon })$ and $\vert A^{\prime\prime}_{\varepsilon }\setminus A^{\prime}_{\varepsilon }\vert = O(\varepsilon )$. Such sets are local minimizers for $F_{\varepsilon }$ and hence the minimizing movement of $F_{\varepsilon }$ from either of them is trivial. As a consequence, if $A_{\varepsilon }(t)$ is a minimizing movement for $F_{\varepsilon }$ from A ₀ we have

$$\displaystyle{A^{\prime}_{\varepsilon } \subset A_{\varepsilon }(t) \subset A^{\prime\prime}_{\varepsilon }.}$$

This shows that for any set A ₀ the only limit $\lim _{\varepsilon \rightarrow 0}A_{\varepsilon }(t)$ of minimizing movements for $F_{\varepsilon }$ from A ₀ is the trivial motion A(t) = A ₀.

We now compute the minimizing movements along the sequence $F_{\varepsilon }$ with initial datum a rectangle, and compare them with the flat flow described in the previous section.

For simplicity of computation we deal with a constrained case, when:

• For every $\varepsilon$ the initial datum $A_{0} = A_{0}^{\varepsilon }$ is a rectangle centered in 0 such that $F_{\varepsilon }(A) = {\mathcal{H}}^{1}(\partial A)$ (i.e., its edge lengths L _0, j belong to $2\varepsilon \mathbb{Z}$). In analogy with x ₀ in the example in Sect. 8.4, if this does not hold then either it does after one iteration or we have a pinned state A _k = A ₀ for all k.

• All competing A are rectangles with $F_{\varepsilon }(A) = {\mathcal{H}}^{1}(\partial A)$ centered in 0. The fact that all competing sets are rectangles follows as for the flat flow in the previous section. The fact that $F_{\varepsilon }(A_{k}) \leq F_{\varepsilon }(A_{k-1})$ then implies that the minimal rectangles satisfy $F_{\varepsilon }(A_{k}) = {\mathcal{H}}^{1}(\partial A_{k})$. The only real assumption at this point is that they are centered in 0. This hypothesis can be removed, upon a slightly more complex computation, which would only make the arguments less clear.

After this simplifications, the incremental problem is exactly as in (9.17) since for competing sets we have $F_{\varepsilon }(A) = F(A)$, the only difference being that now $L_{k,1},L_{k,2} \in 2\varepsilon \mathbb{Z}$. The problem in terms of $\bigtriangleup L_{j}$, using the same simplification for (9.18) as in the previous section, is then

$$\displaystyle\begin{array}{rcl} \min {\Bigl \{2(L_{k,1}+\varDelta L_{1})+2(L_{k,2}+\varDelta L_{2})+ \frac{L_{k,1}} {4\tau } {(\varDelta L_{2})}^{2}+ \frac{L_{k,2}} {4\tau } {(\varDelta L_{1})}^{2}:\varDelta L_{ j} \in 2\varepsilon \mathbb{Z}\Bigr \}}.& &{}\end{array}$$

(9.22)

This is a minimization problem for a parabola as the ones in Sect. 8.4 that gives

$$\displaystyle{ \varDelta L_{1} = -{\Bigl \lfloor \frac{4\tau } {\varepsilon L_{k,2}} + \frac{1} {2}\Bigr \rfloor}\varepsilon \qquad \mbox{ if } \frac{4\tau } {\varepsilon L_{k,2}} + \frac{1} {2}\not\in \mathbb{Z} }$$

(9.23)

(the other cases giving two solutions), and an analogous equation for Δ L ₂. Passing to the limit, we have the system of ODEs governed by the parameter

$$\displaystyle{w =\lim _{\varepsilon \rightarrow 0}\frac{\tau } {\varepsilon }}$$

(which we may suppose exists, up to subsequences), which reads as

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} L_{1}^{\prime} = -\frac{1} {w}{\Bigl \lfloor\frac{4w} {L_{2}} + \frac{1} {2}\Bigr \rfloor} \quad \\ L_{2}^{\prime} = -\frac{1} {w}{\Bigl \lfloor\frac{4w} {L_{1}} + \frac{1} {2}\Bigr \rfloor}.\quad \end{array} \right. }$$

(9.24)

Note that the right-hand side is a discontinuous function of L _j , so some care must be taken at times t when $\frac{4w} {L_{j}(t)} + \frac{1} {2} \in \mathbb{Z}$. However, apart from some exceptional cases, this condition holds only for a countable number of t, and is therefore negligible.

We can compare the resulting minimizing movements with the crystalline curvature flow, related to F.

• (total pinning) if $\tau \ll \varepsilon$ (w = 0) then we have A(t) = A ₀.

• (crystalline curvature flow) if $\varepsilon \ll \tau$ then we have the minimizing movements described in the previous section.

• (partial pinning/asymmetric curvature flow) if 0 < w < +∞ then we have

  • (i) (total pinning) if both L _0, j > 8w then the motion is trivial A(t) = A ₀.

  • (ii) (partial pinning) if L _0, 1 > 8w, L _0, 2 < 8w and $\frac{4w} {L_{0,2}} + \frac{1} {2}\not\in \mathbb{Z}$ then the horizontal edges do not move, but they contract with constant velocity until L ₁(t) = 8w.

  • (iii) (asymmetric curvature flow) if L _0, 1 ≤ 8w and L _0, 2 < 8w then we have a unique motion with A(t) ⊂⊂ A(s) if t > s, up to a finite extinction time. Note, however, that the sets A(s) are not homothetic, except for the trivial case when A ₀ is a square.

Some cases are not considered above, namely those when we do not have uniqueness of minimizers in the incremental problem. This may lead to a multiplicity of minimizing movements, as remarked in Sect. 8.4.

It is worthwhile to highlight that we may rewrite the equations for L′ _j as a variation of the crystalline curvature flow; e.g., for L ₁′ we can write it as

$$\displaystyle{L_{1}^{\prime} = -f{\Bigl (\frac{L_{2}} {w} \Bigr )} \frac{4} {L_{2}},\qquad \text{ with }f(z) = \frac{z} {4}{\Bigl \lfloor\frac{4} {z} + \frac{1} {2}\Bigr \rfloor}.}$$

This suggests that the ‘relevant’ homogenized problem is the one obtained for $\frac{\tau } {\varepsilon }\,=\,1$, as all the others can be obtained from this one by a scaling argument.

We note that the scheme can be applied to the evolution of more general sets, but the analysis of the rectangular case already highlights the new features deriving from the microscopic geometry.

9.6 Flat Flow with Oscillating Forcing Term

We now consider another minimizing-movement scheme linked to the functional F in (9.16). In this case, the oscillations are given by a lower-order forcing term. We consider, in ${\mathbb{R}}^{2}$,

$$\displaystyle{ G_{\varepsilon }(A) =\int _{\partial A}\|\nu \|_{1}d{\mathcal{H}}^{1} +\int _{ A}g{\Bigl (\frac{x_{1}} {\varepsilon } \Bigr )}\mathit{dx}, }$$

(9.25)

where g is 1-periodic and even, given by

$$\displaystyle{g(s) = \left \{\begin{array}{@{}l@{\quad }l@{}} \alpha \quad &\text{if dist}\,(x, \mathbb{Z}) < \frac{1} {4} \\ \beta \quad &\text{otherwise}, \end{array} \right.}$$

with $\alpha,\beta \in \mathbb{R}$ and α <β. Note that the additional term may be negative, so that this functional is not positive; however, the minimizing-movement scheme can be applied unchanged.

Since the additional term converges continuously in L ¹ as $\varepsilon \rightarrow 0$, the Γ-limit is simply

$$\displaystyle{ G(A) =\int _{\partial A}\|\nu \|_{1}d{\mathcal{H}}^{1} + \frac{\alpha +\beta } {2} \vert A\vert. }$$

(9.26)

9.6.1 Flat Flow with Forcing Term

We now consider minimizing movements for G. As in Sect. 9.4.1, we only deal with a constrained problem, when both the initial datum and the competing sets are rectangles centered in 0. With the notation of Sect. 9.4.1, we are led to the minimum problem

$$\displaystyle\begin{array}{rcl} & & \min \Bigl \{2(L_{k,1} +\varDelta L_{1} + L_{k,2} +\varDelta L_{2}) + \frac{L_{k,1}} {4\tau } {(\varDelta L_{2})}^{2} + \frac{L_{k,2}} {4\tau } {(\varDelta L_{1})}^{2} {}\\ & & \quad + \frac{\alpha +\beta } {2} (L_{k,1} +\varDelta L_{1})(L_{k,2} +\varDelta L_{2})\Bigr \}. {}\\ \end{array}$$

The minimizing pair (Δ L ₁, Δ L ₂) satisfies

$$\displaystyle{ \frac{\varDelta L_{1}} {\tau } = -{\Bigl ( \frac{4} {L_{k,2}} + (\alpha +\beta ){\Bigl (1 + \frac{\varDelta L_{2}} {L_{k,2}}\Bigr )}\Bigr )} }$$

(9.27)

and the analogous equation for $\frac{\varDelta L_{2}} {\tau }$. Passing to the limit we have

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} L_{1}^{\prime} = -{\Bigl ( \frac{4} {L_{2}} + \alpha +\beta \Bigr )} \quad \\ L_{2}^{\prime} = -{\Bigl ( \frac{4} {L_{1}} + \alpha +\beta \Bigr )},\quad \end{array} \right. }$$

(9.28)

so that each edge moves with velocity $\frac{2} {L_{2}} + \frac{\alpha +\beta } {2}$, with the convention that it moves inwards if this number is positive, outwards if it is negative.

Note that if α +β ≥ 0 then L ₁ and L ₂ are always decreasing and we have finite-time extinction, while if α +β < 0 then there is an equilibrium for $L_{j} = 4/\vert \alpha +\beta \vert$, and we have expanding rectangles, with an asymptotic velocity of each side of $\vert \alpha +\beta \vert /2$ as the side length diverges.

9.6.2 Homogenization of Forcing Terms

We treat the case $\tau \ll \varepsilon$ only, in which we may highlight new homogenization phenomena. Again, we consider the constrained case when both the initial datum and the competing sets are rectangles centered in 0 and adopt the notation of Sect. 9.4.1. The geometry of the problem is pictured in Fig. 9.2, where the two colors in the background represent the two values of the forcing term.

Fig. 9.2

Rectangle in a layered environment

Taking into account that $\tau \ll \varepsilon$, the incremental minimum problem can be approximated by

$$ \displaystyle\begin{array}{rcl} & & \min \Bigl \{2(L_{k,1} +\varDelta L_{1} + L_{k,2} +\varDelta L_{2}) + \frac{L_{k,1}} {4\tau } {(\varDelta L_{2})}^{2} + \frac{L_{k,2}} {4\tau } {(\varDelta L_{1})}^{2} \\ & & \quad + \frac{\alpha +\beta } {2} L_{k,1}L_{k,2} + \frac{\alpha +\beta } {2} L_{k,1}\varDelta L_{2} + g\Bigl (\frac{L_{k,1}} {2\varepsilon } \Bigr )L_{k,2}\varDelta L_{1}\Bigr \}. {}\end{array}$$

(9.29)

In considering the term $g(L_{k,1}/2\varepsilon )$ we implicitly assume that τ is so small that both $L_{k,1}/2\varepsilon$ and $(L_{k,1} +\varDelta L_{1})/2\varepsilon$ belong to the same interval where g is constant. This can be assumed up to a number of k that is negligible as τ → 0.

For the minimizing pair of (9.29) we have

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} 2 + \frac{L_{k,2}} {2\tau } \varDelta L_{1} + g{\Bigl (\frac{L_{k,1}} {2\varepsilon } \Bigr )}L_{k,2} = 0\quad \\ 2 + \frac{L_{k,1}} {2\tau } \varDelta L_{2} + \frac{\alpha +\beta } {2} L_{k,1} = 0; \quad \end{array} \right. }$$

(9.30)

that is,

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{\varDelta L_{1}} {\tau } = -{\Biggl ( \frac{4} {L_{k,2}} + 2g{\Bigl (\frac{L_{k,1}} {2\varepsilon } \Bigr )}\Biggr )}\quad \\ \frac{\varDelta L_{2}} {\tau } = -{\Bigl ( \frac{4} {L_{k,1}} + (\alpha +\beta )\Bigr )}. \quad \end{array} \right. }$$

(9.31)

This systems shows that the horizontal edges move with velocity $\frac{2} {L_{k,1}} + \frac{\alpha +\beta } {2}$, while the velocity of the vertical edges depends on the location of the edge and is

$$\displaystyle{ \frac{2} {L_{k,2}} + g{\Bigl (\frac{L_{k,1}} {2\varepsilon } \Bigr )}.}$$

We then deduce that the limit velocity for the horizontal edges of length L ₁ is

$$\displaystyle{ \frac{2} {L_{1}} + \frac{\alpha +\beta } {2}. }$$

(9.32)

As for the vertical edges, we have:

• (mesoscopic pinning) if L ₂ is such that

  $$\displaystyle{{\Bigl ( \frac{2} {L_{2}} +\alpha \Bigr )}{\Bigl ( \frac{2} {L_{2}}+\beta \Bigr )} < 0}$$

  then the vertical edge is eventually pinned in the minimizing-movement scheme. This pinning is not due to the equality $L_{k+1,1} = L_{k,1}$ in the incremental problem, but to the fact that the vertical edge moves in contrasting directions depending on the value of g.

• (homogenized velocity) if, on the contrary, the vertical edge length satisfies

  $$\displaystyle{{\Bigl ( \frac{2} {L_{2}} +\alpha \Bigr )}{\Bigl ( \frac{2} {L_{2}}+\beta \Bigr )} > 0}$$

  then we have a limit effective velocity of the vertical edge given by the harmonic mean of the two velocities $\frac{2} {L_{2}} +\alpha$ and $\frac{2} {L_{2}} +\beta$; namely,

  $$\displaystyle{ \frac{(2 +\alpha L_{2})(2 +\beta L_{2})} {L_{2}{\Bigl (2 + \frac{\alpha +\beta } {2} L_{2}\Bigr )}}. }$$

  (9.33)

We examine some cases explicitly.

• (i) Let $\alpha = -\beta$. Then we have

  $$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} L_{2}^{\prime} = - \frac{4} {L_{1}} \quad \\ L_{1}^{\prime} = -2\frac{(2 -\beta L_{2}) \vee 0} {L_{2}} \,;\quad \end{array} \right. }$$

  (9.34)

  i.e., the vertical edges are pinned if their length is larger than 2∕β. In this case, the horizontal edges move inwards with constant velocity $\frac{2} {L_{0,1}}$. In this way the vertical edges shrink with rate $\frac{4} {L_{0,1}}$ until their length is 2∕β. After this, the whole rectangle shrinks in all directions. The stages of this evolution are pictured in Fig. 9.3.

  Fig. 9.3

  

  Stages in the motion according to system (9.34)

• (ii) Let α <β < 0. Then for the vertical edges we have an interval of ‘mesoscopic pinning’ corresponding to

  $$\displaystyle{ \frac{2} {\vert \beta \vert } \leq L_{2} \leq \frac{2} {\vert \alpha \vert } }$$

  (9.35)

  The velocity of the vertical edges in dependence of their length is then given by

  $$\displaystyle{v = \left \{\begin{array}{@{}l@{\quad }l@{}} 0 \quad &\text{if (<InternalRef RefID="Equ37">9.35</InternalRef>) holds} \\ \frac{(2 +\alpha L_{2})(2 +\beta L_{2})} {L_{2}{\bigl (2 + \frac{\alpha +\beta } {2} L_{2}\bigr )}} \quad &\text{otherwise} \end{array} \right.}$$

  and is pictured in Fig. 9.4. Instead, the velocity of the horizontal edges is given by (9.32), so that they move inwards if

  $$\displaystyle{L_{1} < \frac{4} {\vert \alpha +\beta \vert },}$$

  and outwards if $L_{1} > \frac{4} {\vert \alpha +\beta \vert }$.

Fig. 9.4

Velocity of the vertical hedges with an interval of mesoscopic pinning

In this case we can consider as initial datum a square of side length L ₀. We have the following cases:

• If $L_{0} \leq \frac{2} {\vert \beta \vert }$ then all edges move inwards until a finite extinction time.

• If $\frac{2} {\vert \beta \vert } < L_{0} < \frac{4} {\vert \alpha +\beta \vert }$ then first only the horizontal edges move inwards until the vertical edge reaches the length $\frac{2} {\vert \beta \vert }$, after which all edges move inwards.

• If $\frac{4} {\vert \alpha +\beta \vert } < L_{0} < \frac{2} {\vert \alpha \vert }$ then first only the horizontal edges move outwards until the vertical edge reaches the length $\frac{2} {\vert \alpha \vert }$, after which all edges move outwards.

• If $L_{0} \geq \frac{2} {\vert \alpha \vert }$ then all edges move outwards, and the motion is defined for all times. The asymptotic velocity of the vertical edges as the length of the edges diverges is

  $$\displaystyle{{\Bigl | \frac{2\alpha \beta } {\alpha +\beta }\Bigr |},}$$

  lower than ${\Bigl |\frac{\alpha +\beta } {2} \Bigr |}$ (the asymptotic velocity for the horizontal edges).

The critical case can be shown to be $\varepsilon \sim \tau$, so that for $\varepsilon \ll \tau$ we have the flat flow with averaged forcing term described in Sect. 9.6.1. The actual description in the case $\varepsilon \sim \tau$ would involve a complicated homogenization argument for the computation of the averaged velocity of vertical sides.

Appendix

The variational approach for motion by mean curvature is due to Almgren et al. [2]. The variational approach for crystalline curvature flow is contained in a paper by Almgren and Taylor [1].

The homogenization of the flat flow essentially follows the discrete analog contained in the paper by Braides et al. [3]. In that paper more effects of the microscopic geometry are described for more general initial sets. The homogenization with forcing term is part of ongoing work with A. Malusa and M. Novaga.

Geometric motions with a non-trivial homogenized velocity are described in the paper by Braides and Scilla [4], where example are shown of geometries which do not influence the crystalline perimeter obtained as Γ-limit, but do influence various features of the evolution.