Universit`
a di Pisa
DIPARTIMENTO DI MATEMATICA Corso di Laurea in Matematica
Tesi di laurea magistrale
Mean Curvature Motion as a Curve of Maximal Slope
-The radial case
Candidato
Nicola Picenni
Relatore
Contents
1 Introduction 1
1.1 The Mean Curvature Motion . . . 1
1.2 Maximal Slope Curves . . . 3
1.3 Modica-Mortola functional and Allen-Cahn equation . . . 5
1.3.1 Allen-Cahn solutions as curves of maximal slope . . . 6
1.4 Structure of the thesis . . . 7
2 Mean curvature motion as a curve of maximal slope 9 3 The Modica-Mortola functionals 19 3.1 Γ-convergence . . . 19
3.2 Metric slope . . . 21
3.2.1 Metric slope in radial symmetry . . . 25
4 Γ-liminf of metric slopes 29
5 Convergence of Gradient-Flows 41
1
Introduction
1.1
The Mean Curvature Motion
The mean curvature motion in Rnis a geometric evolution problem that has been extens-ively studied since the 80s with different methods. In the smooth case it is possible to prove short time existence for the evolution in every dimension and codimension. This can be done, for example, by parametrizing both the initial datum and the evolving surface with a fixed manifold. In this way the problem is reduced to solving a system of partial differential equations on a given smooth manifold. In the special case of smooth curves in the plane, one can also prove existence up to the so-called “extinction time”, when the curve degenerates to a point and then “disappears”.
On the other hand, in dimension n ≥ 3 it is well known that the evolution can develop singularities (different from the extinction) in finite time, even when the initial datum is a smooth hypersurface. After these singularities the moving hypersurface can change its topology so that it is not possible to continue the flow using the same parameter space. In order to overcome this problem, and also to extend the flow to nonsmooth objects, many weak definitions of mean curvature flow have been proposed. Depending on the weak notion of surface, curvature, and evolution, we can classify these approaches into four main categories as follows.
• In the measure theoretical definition proposed by Brakke in [10] the moving objects are varifolds, and both the curvature and the evolution equation are defined via integration by parts, in analogy with the usual definition of Sobolev spaces. Not-ably, one of the equalities that holds true in the smooth case becomes an inequality in this weak definition.
• In the level set method, developed independently (in codimension one) by Chen, Giga and Goto in [17] and by Evans and Spruck in [11], the evolving objects are level sets of a time dependent function, which is a viscosity solution to a particular PDE that, roughly speaking, says that all level sets evolve at the same time according to their mean curvature. This method has been extended to any codimension by Ambrosio and Soner in [3].
• In the variational approach developed by Almgren, Taylor and Wang in [1] the evolution is defined as the limit of some sort of backward Euler scheme, in which every new step is computed by minimizing the perimeter functional with a penal-ization depending on a non symmetric “distance”. This method fits in what is now called the theory of minimizing movements by De Giorgi.
• In the method of barriers, introduced by De Giorgi and developed by Bellettini and Paolini in [8], the flow of any object is approximated from outside and from inside by smooth evolutions of smooth hypersurfaces.
Each of these weak definitions has its own advantages and drawbacks, and the rela-tions among them have been deeply investigated. The general idea is that the level set approach and the barriers method are equivalent and deliver some sort of “maximal solu-tion” that contains all the evolutions provided by the other approaches (see [6, 7, 19]). On the other hand, even if we can find many statements and a huge literature in this direction, some steps are still open or at least not fully understood.
In the same years some approximations or regularizations of the mean curvature flow have been introduced. Let us mention three of them.
• The Allen-Cahn equation, which is the gradient flow of the Modica-Mortola func-tional. It is well known that the Modica-Mortola functional Γ-converges to the perimeter functional. On the other hand, the mean curvature is the “gradient” of the perimeter functional, at least in the sense that the inner variation of the peri-meter computed in a smooth set along a smooth vector-field is the scalar product (on the boundary) of the mean curvature with the vector-field. As a consequence, it is reasonable to expect that the gradient flow of the Modica-Mortola functional converges in some sense to the mean curvature motion. As we see in the sequel, this is true only up to a time rescaling.
• The so-called thresholding scheme proposed by Merriman, Bence and Osher in [21], where the mean curvature motion is approximated by alternating steps in which the heat equation is solved starting from a characteristic function and steps where the solution is set again equal to 0 or 1 when it is lower or greater than 1/2. • The higher order regularization proposed by De Giorgi in [13] and then considered
by Mantegazza in [20] (see also [15] and [4]), where the mean curvature motion is approximated by the gradient flow of a functional involving both the perimeter and the norm of the curvature and its derivatives up to a suitable order.
It would be nice to have a unifying framework for all these approximations. In this direction, important results have been obtained through the theory of viscosity solutions, which is quite robust when passing to the limit. For example, in this framework it was proved that solutions to the Allen-Cahn equation converge in a suitable sense to the mean curvature motion in the level set formulation (see [16]). On the other hand, this approach depends strongly on the maximum principle, and for this reason it is probably hopeless when the maximum principle does not hold, for example in the case of the approximation with higher order derivatives.
As an alternative, one could try to fit the mean curvature motion in the general theory of maximal slope curves in metric spaces (see the next section for a precise definition). This theory provides a notion of gradient flow that involves only integrals and inequalities, which are more stable than derivatives and equalities when passing to the limit.
Therefore, it would be desirable to characterize the mean curvature motion as the gradient flow of some functional (for example the perimeter or some suitable extension to
1.2. Maximal Slope Curves
a more general ambient space) with respect to a suitable distance in the ambient space, and then deduce the convergence of the approximating problems from general abstract convergence results for maximal slope curves. This would also clarify more formally in which sense “the mean curvature motion is the gradient flow of the perimeter”.
Some results in the literature (see [27]) seem to point in this direction, at least in the case of the Allen-Cahn equation, but a clear statement of the mean curvature motion as maximal slope curve seems to be still missing.
In the thesis we investigate the possibility to develop such a theory at least in the toy model of the radially symmetric case and to prove the convergence of the radial Allen-Cahn approximation within this setting.
1.2
Maximal Slope Curves
The theory of maximal slope curves was introduced by De Giorgi, Marino and Tosques in [14] to give a definition of gradient flow for functionals defined on metric spaces. Here we follow the modern presentation given in [9] (see also the book [2] for a complete theory of metric gradient flows).
Let (X, d) be a metric space, let F : X → R be a function and let u : [a, b] → X be a curve. We can now give some definitions.
Definition 1.2.1 (Descending metric slope). Let x ∈ X be a non-isolated point such
that F (x) ∈ R. We define the descending metric slope of F at the point x as |∇F |(x) := lim sup
y→x
max{F (x) − F (y), 0}
d(x, y) .
We notice that if X is a Hilbert space and F ∈ C1(X), then the slope of F is the norm of its gradient.
Definition 1.2.2 (Metric derivative). We say that u ∈ AC2([a, b], X) if there exists a
function g ∈ L2([a, b], [0, +∞)) such that
d(u(t), u(s)) ≤
ˆ t s
g(τ )dτ ∀a ≤ s < t ≤ b.
In this case, the smallest function g with this property is said to be the metric derivative of u and is denoted by | ˙u|(t).
It is possible to prove that
| ˙u|(t) = lim
h→0
d(u(t + h), u(t))
|h| for L
1-a.e. t ∈ [a, b].
Moreover, if X is a Hilbert space, then
u ∈ AC2([a, b], X) ⇐⇒ u ∈ H1([a, b], X) and the metric derivative of u is the L2-norm of its weak derivative.
Definition 1.2.3 (Curve of maximal slope). A curve u ∈ AC2([a, b], X) is said to be a
curve of maximal slope for a function F : X → R if there exists a nonincreasing function
ϕ : [a, b] → R such that ϕ(t) = F (u(t)) for almost every t ∈ [a, b] and ϕ(s) − ϕ(t) ≥ 1 2 ˆ t s |∇F |(u(τ ))2dτ +1 2 ˆ t s | ˙u|(τ )2dτ ∀a ≤ s < t ≤ b.
Curves of maximal slope are a natural generalization of gradient flows that can be defined in the metric setting. It is easy to see that for a curve of maximal slope the equality | ˙u|(t) = |∇F |(u(t)) holds for almost every t. However, the definition of curve of
maximal slope is much stronger than this equality, since it also takes into account the direction of the curve, as it happens for classical gradient flows in Hilbert spaces.
Indeed, if X is a Hilbert space and F ∈ C1(X), then u is a curve of maximal slope for F if and only if it solves the classical gradient flow equation
˙
u(t) = −∇F (u(t)).
Moreover, as for classical gradient flows, we have the following (global) existence result.
Theorem 1.2.4. Let F : X → R be a function satisfying the following assumptions:
• inf F > −∞,
• F is locally coercive and lower semicontinuous,
• For every converging sequence xn → x∞ such that supnF (xn) + |∇F |(xn) < +∞
it turns out that F (xn) → F (x∞) and lim infn→+∞|∇F |(xn) ≥ |∇F |(x∞).
Then for every u0 ∈ X such that F (u0) ∈ R there exists u ∈ AC2([0, +∞), X) with u(0) = u0 that is a curve of maximal slope for F .
This theorem can be proved using a minimizing movement starting at u0, namely
discrete paths that locally minimize the values of F . Actually, some of the assumptions on F can be relaxed, but this is not so important for our purpose.
The fundamental feature of maximal slope curves is their stability with respect to Γ-convergence of functionals and slopes. A possible result in this direction is the following.
Theorem 1.2.5. Let (Xε, dε) and (X, d) be metric spaces and let us assume that there
exists a notion of convergence * for which we can say that a family of points xε ∈ Xε converges to a point x ∈ X.
Let Fε : Xε → [0, +∞] and F : X → [0, +∞] be non-negative functionals and let uε ∈ AC2([a, b], Xε) be curves of maximal slope for Fε. Let ϕε be the nonincreasing
functions given by the definition of curves of maximal slope. Let us assume that the following hypotheses hold
• There exists a curve u : [a, b] → X such that uε(t) * u(t) for every t ∈ [a, b],
• sup
ε>0
1.3. Modica-Mortola functional and Allen-Cahn equation
• For every sequence of real numbers εm→ 0 and every sequence of points uεm ∈ Xεm
such that uεm * u for some u ∈ X and supmFεm(uεm) + |∇Fεm|(uεm) < +∞ it
turns out that
lim
m→+∞Fεm(uεm) = F (u) and lim infm→+∞|∇Fεm|(uεm) ≥ |∇F |(u).
• lim inf ε→0 ˆ t s | ˙uε|(τ )2dτ ≥ ˆ t s
| ˙u|(τ )2dτ for every a ≤ s < t ≤ b,
where |∇Fε|(uε(t)) is the slope of Fε in (Xε, dε) and | ˙uε|(τ ) is the metric derivative of
uε in (Xε, dε). Then it turns out that u is a curve of maximal slope for F in (X, d).
Proof. Since ϕε are nonincreasing, by Helly’s lemma there exist a subsequence εm → 0
and a nonincreasing function ϕ : [a, b] → R such that ϕεm(t) → ϕ(t) for every t ∈ [a, b].
We know that for every m ∈ N we have that
M ≥ ϕεm(a) − ϕεm(b) ≥ 1 2 ˆ b a |∇Fεm|(uεm(t))2dt +1 2 ˆ b a | ˙uεm|(t)2dt.
Therefore, if we compute the liminf, by Fatou’s lemma we deduce that ˆ b a lim inf m→+∞|∇Fεm|(uεm(t)) 2dt ≤ lim inf m→+∞ ˆ b a |∇Fεm|(uεm(t))2dt ≤ 2M.
Moreover, there exists a set E ⊂ [a, b] with Lebesgue measure equal to zero such that
Fεm(t) = ϕεm(t) ≤ M for every t ∈ [a, b] \ E and for every m ∈ N.
Therefore, for almost every t ∈ [a, b], there exists a subsequence εmk → 0 (depending
on t) such that supmFεmk(uεmk(t)) + |∇Fεmk|(uεmk(t)) < +∞, that is the subsequence
on which the liminf of |∇Fεm|(uεm(t)) is attained as a limit.
By the third assumption we deduce that for almost every t ∈ [a, b] we have that
lim
k→+∞Fεmk(uεmk(t)) = F (u(t)) and m→+∞lim inf|∇Fεm|(uεm(t)) ≥ |∇F |(u(t)).
This means that F (u(t)) = ϕ(u(t)) for almost every t ∈ [a, b]. Then, exploiting also the fourth assumption we deduce that
ϕ(s) − ϕ(t) ≥ 1 2 ˆ t s lim inf m→+∞|∇Fεm|(uεm(t)) 2dt +1 2m→+∞lim inf ˆ t s | ˙uεm|(t) 2dt ≥ 1 2 ˆ t s |∇F |(u(t))2dt +1 2 ˆ t s | ˙u|(t)2dt
for every s, t ∈ [a, b] with s < t. This means that u is a maximal slope curve for F .
1.3
Modica-Mortola functional and Allen-Cahn equation
We now give an informal introduction to the Modica-Modica functional and its gradient flow, in order to explain the motivation of our analysis. More precise statements are given in the next chapters.
The Modica-Mortola functional is defined for every ε > 0 as M Mε(u) := ˆ Rn ε|∇u(x)|2+1 εW (u(x)) dx,
where W is a so-called double well potential, for example W (u) = (1 − u2)2. The
L2-gradient of this functional turns out to be (at least formally)
∇M Mε(u) = −2ε∆u +
1
εW 0
(u).
In 1977 Modica and Mortola (see [23, 24]) proved (with a slightly different potential
W ) that there exists a constant σ > 0 such that
Γ − lim ε→0M Mε(u) = σ Per({u = −1}) if u ∈ BVloc(Rn, {±1}), +∞ otherwise.
The Allen-Cahn equation is the gradient flow of the Modica-Mortola functional, rescaled by a factor 1/ε. This leads to the semilinear parabolic equation
∂tuε= 2∆uε−
1
ε2W 0(u
ε).
The factor 1/ε produces a speed-up effect in the evolution, yielding a non trivial limit as ε → 0. This limit has been investigated in many papers. Here for the sake of shortness we quote just two of them, pointing in different directions.
• Evans, Soner and Souganidis in [16] proved that viscosity solutions to the Allen-Cahn equation converge to ±1 almost everywhere, and the limit interface between the two values is contained in the level set mean curvature flow. This result requires a special choice of well-prepared initial data.
• Ilmanen proved in [18] that the energy density
µεt := ε|∇uε|2+ 1 εW (uε) · Ln
of a family uε of smooth solutions of the Allen-Cahn equation converges to a varifold moving in the sense of Brakke. Also this result requires well-prepared initial data.
1.3.1 Allen-Cahn solutions as curves of maximal slope
Let us first consider the non rescaled gradient flow of the Modica-Mortola functional, that is ∂tuε= 2ε∆uε− 1 εW 0 (uε).
It is easily seen that the solutions of this equation can be characterized as curves of maximal slope of M Mε in L2(Rn). However, this is not useful to approximate the
1.4. Structure of the thesis
ε → 0. This fact is in accord with Theorem 1.2.5, because the L2-slope of the perimeter functional is null (at least on smooth sets).
Similarly, if one tries to consider the gradient flow of the perimeter functional with respect to the L1-distance, it turns out that it is different from the mean curvature flow. This can be seen by considering two disjoint balls with different radii and noticing that the L1 gradient flow shrinks only the smallest one.
Actually it is well known that the right equation to consider in order to approximate the mean curvature flow is the Allen-Cahn equation. The solutions of this equation turn out to be the gradient flow of M Mεin L2loc(Rn) with respect to the distance dε:= ε
1 2dL2,
and therefore they are characterized by the following inequality
M Mε(uε(s)) − M Mε(uε(t)) ≥ 1 2ε ˆ t s k∇M Mε(uε(τ ))k2L2 dτ + ε 2 ˆ t s k∂tuε(τ )k2L2 dτ.
Indeed, it turns out that the metric slope of M Mε with respect to dε is given by |∇M Mε|(u) = ε−12k∇M Mε(u)kL2, while the metric derivative of a curve uεwith respect
to dε is given by | ˙uε|(t) = ε 1
2k∂tu(t)k L2.
At this point, one would like to compute the limit of each term in the previous inequality and find a similar inequality for the limit of the solutions. The next step would be finding a distance on a space of subsets of Rn for which the first term in the right-hand side of the limit is the integral of the metric slope of the perimeter functional and the second term is the integral of the metric derivative of the limit evolution.
The ultimate goal would be to set this result in the general framework of some abstract result in the spirit of Theorem 1.2.5, finding a notion of convergence from (L2loc(Rn), dε) to this new metric space and proving that the convergence of the terms in
the previous inequality corresponds to the hypotheses of Theorem 1.2.5.
This is more or less what Serfaty did in [27], relying on results by R¨oger and Sch¨atzle in [26], who computed the Γ-limit of |∇M Mε|(u), and by Mugnai and R¨oger in [25], who proved that | ˙uε| converges in a suitable sense to a generalized velocity. However, these
results have been proved only when the space dimension is small (n ≤ 3) and the limit evolution was only proved to be a Brakke’s flow. In particular, it was not proved to be a curve of maximal slope in a metric space, since Brakke’s generalized velocity is not a metric derivative.
1.4
Structure of the thesis
In the thesis we try to pursue the path that we have just outlined, under the additional assumption of radial symmetry, but without any restriction on the space dimension. Moreover, in this simplified setting, we try to improve the results on the limit evolution, showing that it is a curve of maximal slope in an appropriate setting. More precisely this thesis is organized as follows.
In Chapter 2 we introduce a distance on (n − 1)-dimensional radial “subsets” of Rn in such a way that the curves of maximal slope of the area functional are motion by mean curvature.
In Chapter 3 we introduce the Modica-Mortola functional, we give a proof of the Γ-convergence to the area functional and we provide a rigorous computation of the metric
slope that gives also some regularity information on functions with finite slope, without the radiality assumption. Then we deduce a particular formula for the slope in the case of radial symmetry.
In Chapter 4 we prove a Γ-liminf inequality for the slope of the Modica-Mortola functional, under the assumption of boundedness of the energy and radial symmetry. Moreover, we show that sequences with bounded slope are recovery sequences for the Modica-Mortola functional, provided that we use a good definition of the limit functional (that is the area of interfaces) that takes into account multiplicities. These are the same results that R¨oger and Sch¨atzle proved in [26] without the radiality assumption, when the space dimension is n ≤ 3.
Finally, in Chapter 5, we address the problem of the convergence of the gradient flows, explaining what are the main difficulties in completing the proof of such a statement. Moreover, we describe a counterexample that shows that the problem is actually quite delicate, and some standard arguments are too rough to be successfully applied in this situation.
2
Mean curvature motion as a
curve of maximal slope
The aim of this chapter is to provide a characterization of the evolutions by mean curvature of radially symmetric “objects” in Rn (n ≥ 2) as the curves of maximal slope of a functional in a suitable metric space.
Since we would like to exploit this characterization to describe the limit of the inter-faces that arise from the solutions of the Allen-Cahn equation, we would like to allow these “objects” to have some multiplicities. Indeed, we see in the following chapters that these interfaces can accumulate one on another, so that the limit interfaces can be made of several overlapping surfaces. The appropriate “objects” to describe this phenomenon turn out to be (n − 1)-dimensional integral varifolds on Rn.
However, in our simplified case of radial symmetry, integral varifolds are simply linear combinations of surface measures of spheres centred in the origin with positive integer coefficients that encode multiplicities.
Hence we can parametrize these varifolds with monotone sequences of non-negative real numbers, corresponding to the radii of the spheres, where equal values stand for a sphere with multiplicity. Moreover, we consider only families of spheres with finite total area. In the end, we deal with the following space
Rn:=
(
(ri)i∈N+ : ri ≥ ri+1≥ 0 ∀i ∈ N+,
+∞ X i=1 rn−1i < +∞ ) ,
where we identify a sequence (ri) with the measureP
iHn−1b∂Bri.
We endow this space with the following distance
dn((ri), (si)) := 2√ωn−1 n + 1 v u u t +∞ X i=1 r n+1 2 i − s n+1 2 i 2 .
Remark 2.0.1. (Rn, dn) is a metric space. Indeed, dn is clearly symmetric and non
de-generate. As for the triangular inequality, it is a consequence of the triangular inequality in the space `2. In fact, (Rn, dn) is actually isometric to a subspace of `2 through the
map (ri) 7→ r n+1 2 i .
However, the linear structure thatRn inherits in this way is quite artificial, since there
radius. In fact, varifolds already have a linear structure that is clearly different from this one. For this reason we avoid using this property ofRn in the sequel.
Given (ri) ∈Rn we can define its total mass, that is given by A((ri)) := +∞ X i=1 Hn−1b∂Bri(Rn) = +∞ X i=1 ωn−1rin−1.
From now on, we call this quantity the area functional, since it coincides with the area of the support of the measure, counted with multiplicity.
We can now compute the descending metric slope of the area functional in (Rn, dn),
whose square turns out to be a multiple of the Willmore functional, that is the integral on a surface of the square of its mean curvature.
Lemma 2.0.2. Let A :Rn → [0, +∞) be the area functional. Let (ri) ∈Rn and let us set N ((ri)) := sup{i ∈ N+: ri> 0} ∈ N+∪ {+∞}. Then it turns out that
|∇A|((ri)) = (n − 1)√ωn−1 v u u u t N ((ri)) X i=1 rn−3i . Proof. We first prove that
|∇A|((ri)) ≤ (n − 1) √ ωn−1 v u u u t N ((ri)) X i=1 rn−3i . (2.0.1) To this end, we can assume that the right-hand side is finite. Therefore, fixing ε > 0, there exists M ∈ N+ such that
N ((ri))
X
i=M
rin−3< ε2. (2.0.2) Indeed, if N ((ri)) = +∞, then the existence of such M follows from the finiteness of
the series. Otherwise, it is enough to choose M = N ((ri)) + 1.
By definition of descending metric slope and elementary algebra we have that
|∇A|((ri)) = lim sup
(si)→(ri) (A((ri)) − A((si)))+ dn((ri), (si)) = lim sup (si)→(ri) (n + 1)√ωn−1 2 P i rn−1i − s n−1 i s P i r n+1 2 i − s n+1 2 i 2 ≤ lim sup (si)→(ri) (n + 1)√ωn−1 2 P i<M rn−1i − s n−1 i s P i<M r n+1 2 i − s n+1 2 i 2 + lim sup (si)→(ri) (n + 1)√ωn−1 2 P i≥M rn−1i − sn−1i s P i≥M r n+1 2 i − s n+1 2 i 2 .
We observe that we can forget the positive part in the definition of descending metric slope since we can assume without loss of generality that si ≤ ri for every i ∈ N+.
In order to estimate the numerators we notice that by the convexity of the function
t 7→ tn−1 and Cauchy-Schwarz inequality we deduce that for every I ⊆ {1, . . . , N ((ri))}
we have X i∈I rn−1i −sn−1i ≤X i∈I (n−1)rin−2(ri−si) ≤ (n−1) s X i∈I rn−3i · s X i∈I rn−1i (ri− si)2. (2.0.3)
As for the denominators, we treat separately the term with a finite sum and the term with the tail of the series. For the first one, we use a first order Taylor expansion to deduce that v u u t X i<M r n+1 2 i − s n+1 2 i 2 = n + 1 2 s X i<M rin−1(ri− si)2+ o((r i− si)2). It follows that lim sup (si)→(ri) (n + 1)√ωn−1 2 P i<M rn−1i − s n−1 i s P i<M r n+1 2 i − s n+1 2 i 2 ≤ lim sup (si)→(ri) (n − 1)√ωn−1 q P i<M rin−3 q P i<M rn−1i (ri− si)2 q P i<M rin−1(ri− si)2+ o((ri− si)2) = (n − 1)√ωn−1 s X i<M rin−3.
Concerning the term with the tail of the series, we observe that for every couple of non-negative real numbers s ≤ r we have that
rn−1(r − s)2 =rn−12 (r − s) 2 =rn+12 − r n−1 2 s 2 ≤rn+12 − s n+1 2 2 .
Therefore, recalling (2.0.3) and (2.0.2), we deduce that
lim sup (si)→(ri) (n + 1)√ωn−1 2 P i≥M rn−1i − sn−1i s P i≥M r n+1 2 i − s n+1 2 i 2 ≤ lim sup (si)→(ri) (n − 1)(n + 1)√ωn−1 2 q P i≥M rn−3i q P i≥M rin−1(ri− si)2 s P i≥M r n+1 2 i − s n+1 2 i 2 ≤ (n − 1)(n + 1) √ ωn−1 2 s X i≥M rin−3 ≤ (n − 1)(n + 1)ωn−1 2 ε.
Hence, we have proved that |∇A|((ri)) ≤ (n − 1)√ωn−1 s X i<M rn−3i + (n − 1)(n + 1)ωn−1 2 ε ≤ (n − 1)√ωn−1 v u u u t N ((ri)) X i=1 rin−3+(n − 1)(n + 1)ωn−1 2 ε.
Since ε is arbitrarily small, we have proved (2.0.1).
As for the opposite inequality, let us fix M ∈ {1, . . . , N ((ri))}, h > 0 and let (¯si) ∈Rn
be defined by ¯ si := ( ri−rhi if i ≤ M, ri otherwise.
It is clear that (¯si) → (ri) when h → 0. Consequently,
|∇A|((ri)) ≥ lim sup h→0 (n + 1)√ωn−1 2 P i≤M rin−1− ri− rhi n−1 s P i≤M r n+1 2 i − ri−rhi n+1 2 2 .
By a first order Taylor expansion we deduce that
|∇A|((ri)) ≥ lim sup h→0 (n + 1)√ωn−1 2 P i≤M(n − 1)rin−3h + o(h) n+1 2 q P i≤Mrin−3h2+ o(h2) = (n − 1)√ωn−1 s X i≤M rn−3i .
Since we can choose M = N ((ri)) if N ((ri)) is finite, or we can choose M arbitrarily large if N ((ri)) = +∞, then we have proved also the other inequality, so we have
computed the exact value of |∇A|.
In the sequel, in order to simplify the notation, we simply write
|∇A|((ri)) = (n − 1) √ ωn−1 v u u t +∞ X i=1 rin−3,
with the convention that rin−3 = 0 if ri = 0, even if the space dimension n is two or
three.
The next step in the study of the curves of maximal slope of A is the following characterization of absolutely continuous curves inRn.
Lemma 2.0.3. Let u ∈ AC2([a, b],Rn) be an absolutely continuous curve with
square-integrable metric derivative. Then for each component ui of u it turns out that
u
n+1
2 i ∈ H
Moreover the metric derivative of u turns out to be | ˙u|(t) = 2 √ ωn−1 n + 1 v u u t +∞ X i=1 ui(t) n+1 2 02 .
Proof. By the definition of the space AC2([a, b],Rn), for every i ∈ N we have that 2√ωn−1 n + 1 ui(t + h) n+1 2 − ui(t) n+1 2 ≤ dn(u(t + h), u(t)) ≤ ˆ t+h t | ˙u|(s) ds.
A simple application of H¨older’s inequality and Fubini’s theorem leads to
ui(t + h) n+1 2 − ui(t) n+1 2 h L2 ≤ n + 1 2√ωn−1 k| ˙u|kL2,
and this is equivalent to
u
n+1
2
i ∈ H1([a, b], [0, +∞)).
We can now compute the metric derivative. Let us fix M ∈ N and let t ∈ [a, b] be a point such that u(n+1)/2i is differentiable at t for every i ∈ N (this holds for almost every
t). Then
| ˙u|(t)2 = lim sup
h→0 d(u(t + h), u(t))2 h2 ≥ lim sup h→0 4ωn−1 (n + 1)2 M X i=1 1 h2 ui(t + h) n+1 2 − ui(t) n+1 2 2 = 4ωn−1 (n + 1)2 M X i=1 ui(t) n+1 2 02 .
Since M can be chosen arbitrarily large, this proves one inequality. As for the other one, since we have already proved that the series is bounded from above by the metric derivative, it converges to an L2 function. Let t ∈ [a, b] be a Lebesgue point for the series. Then
| ˙u|(t)2= lim sup
h→0 dn(u(t + h), u(t))2 h2 = lim sup h→0 4ωn−1 (n + 1)2 +∞ X i=1 1 h2 ui(t + h) n+1 2 − ui(t) n+1 2 2 = lim sup h→0 4ωn−1 (n + 1)2 +∞ X i=1 1 h2 ˆ t+h t ui(τ ) n+1 2 0 dτ 2 ≤ lim sup h→0 4ωn−1 (n + 1)2 +∞ X i=1 1 |h| ˆ t+h t ui(τ ) n+1 2 02 dτ = lim sup h→0 4ωn−1 (n + 1)2 1 |h| ˆ t+h t +∞ X i=1 ui(τ ) n+1 2 02 dτ = 4ωn−1 (n + 1)2 +∞ X i=0 ui(t) n+1 2 02 ,
where we have used the weak differentiability of u(n+1)/2i , Cauchy-Schwarz inequality, the monotone convergence theorem and the fact that t is a Lebesgue point for the series. This proves the other inequality.
Remark 2.0.4. In general, if u ∈ AC2([a, b],Rn) we can not deduce that its components
ui belong to H1([a, b]).
We can now prove the main result of this chapter, that is the characterization of the radial mean curvature motion as a curve of maximal slope in (Rn, dn).
Theorem 2.0.5. Let u ∈ AC2([0, T ),Rn) be a curve of maximal slope for A. Then for
every i ∈ N and for every t ∈ [0, T ) it turns out that ui(t) =
q
(ui(0)2− 2(n − 1)t)+.
Proof. By the definition of curve of maximal slope we know that there exists a set of
null measure E ⊂ [0, T ) such that for every s, t ∈ [0, T ) \ E with s < t we have that
A(u(s)) − A(u(t)) ≥ 1 2 ˆ t s |∇A|(u(τ ))2dτ + 1 2 ˆ t s | ˙u|(τ )2dτ.
By Lemma 2.0.2 and Lemma 2.0.3 this means that
A(u(s)) − A(u(t)) ≥ (n − 1) 2ω n−1 2 ˆ t s +∞ X i=1 ui(τ )n−3dτ + 2ωn−1 (n + 1)2 ˆ t s +∞ X i=1 ui(τ ) n+1 2 02 dτ.
Let us fix t ∈ [0, T ) \ E and let us set Jt := {i ∈ N+ : ui(t) 6= 0}. We now treat
separately the cases n = 2 and n > 2.
Case 1 (n = 2): In this case, we immediately deduce that Jtis finite for almost every
t ∈ [0, T ), because of the almost everywhere finiteness of the series that appears in the
computation of the slope of A. Let us set
ht:= sup {h > 0 : ui(τ ) 6= 0 ∀i ∈ Jt ∀τ ∈ [t, t + h)} .
Since Jt is finite and ui is continuous for every i ∈ N+, it follows that ht is strictly
positive. Then, for every h < ht such that t + h /∈ E, we have that A(u(t)) − A(u(t + h)) ≥ ω1 2 ˆ t+h t X i∈Jt ui(τ )−1dτ + 2ω1 32 ˆ t+h t X i∈Jt ui(τ ) 3 2 02 dτ.
On the other hand, by Lemma 2.0.3 we know that
u
3 2 i ∈ H
1([a, b], [0, +∞))
for every i ∈ N. Since the function u 7→ u23 is smooth on every interval that does not
i ∈ Jt and every h < ht. Therefore we can exploit the weak differentiability of ui to deduce that A(u(t)) − A(u(t + h)) = ω1 +∞ X i=1 (ui(t) − ui(t + h)) ≤ ω1 X i∈Jt (ui(t) − ui(t + h)) = ω1 X i∈Jt ˆ t+h t − ˙ui(τ ) dτ.
Putting together the upper and the lower estimates on A(u(t)) − A(u(t + h)) and using again the weak differentiability of ui we deduce that
ω1X i∈Jt ˆ t+h t − ˙ui(τ )dτ ≥ ω1 2 ˆ t+h t X i∈Jt ui(τ )−1dτ + 2ω1 32 ˆ t+h t X i∈Jt ui(τ ) 3 2 02 dτ ≥ ω1 2 ˆ t+h t X i∈Jt ui(τ )−1dτ + ω1 2 ˆ t+h t X i∈Jt ui(τ ) ˙ui(τ )2dτ, that is equivalent to ˆ t+h t X i∈Jt ui(τ )− 1 2 + ui(τ ) 1 2u˙i(τ ) 2 dτ ≤ 0.
Therefore for almost every τ ∈ (t, t + ht) and for every i ∈ Jt we have that
ui(τ )− 1
2 + ui(τ ) 1
2u˙i(τ ) = 0.
Since all the functions ui are continuous and strictly positive in (t, t + ht), then we deduce that
˙
ui(τ ) = −
1
ui(τ )
and that ui∈ C∞(t, t + h), so the previous equality holds for every τ ∈ (t, t + h
t).
Hence, we have proved that for every i ∈ N the previous differential equation holds for every τ for which ui(τ ) 6= 0. This easily leads to
ui(t) =
q
(ui(0)2− 2τ )+.
Case 2 (n > 2): In this case, since n − 1 ≥ (n + 1)/2, we have that
u n+1 2 i ∈ H1([a, b], [0, +∞)) ⇒ un−1i ∈ H1([a, b], [0, +∞)) and un−1i 0 = u n+1 2 i 2(n−1)n+1 0 = 2(n − 1) n + 1 u n−3 2 un+12 0 .
Therefore, for every s, t ∈ [0, T ) \ E such that s < t we have that A(u(s)) − A(u(t)) = ωn−1 +∞ X i=1 ui(s)n−1− ui(t)n−1 = 2(n − 1) n + 1 ωn−1 +∞ X i=1 ˆ t s −ui(τ ) n−3 2 ui(τ ) n+1 2 0 dτ,
from which, using the inequality that defines the curves of maximal slope and the mono-tone convergence theorem, we obtain that
2(n − 1) n + 1 ωn−1 +∞ X i=1 ˆ t s −ui(τ ) n−3 2 ui(τ ) n+1 2 0 dτ ≥ (n − 1) 2ω n−1 2 ˆ t s +∞ X i=1 ui(τ )n−3dτ + 2ωn−1 (n + 1)2 ˆ t s +∞ X i=1 ui(τ ) n+1 2 02 dτ = (n − 1) 2ω n−1 2 +∞ X i=1 ˆ t s ui(τ )n−3dτ + 2ωn−1 (n + 1)2 +∞ X i=1 ˆ t s ui(τ ) n+1 2 02 dτ, that is equivalent to +∞ X i=1 ˆ t s (n − 1)ui(τ ) n−3 2 + 2 n + 1 ui(τ ) n+1 2 02 dτ ≤ 0.
Therefore, since all the functions ui are continuous, for every τ ∈ [0, T ) and for every
i ∈ N we have that ui(τ ) n+1 2 0 = −n + 1 2 (n − 1)ui(τ ) n−3 2 = −n + 1 2 (n − 1) ui(τ ) n+1 2 n−3n+1 .
Solving this equation we find
ui(τ ) n+1 2 = u(0)n+12 n+14 − 2(n − 1)τ + !n+14 , that means ui(τ ) = q (u(0)2− 2(n − 1)τ ) +.
It would be very nice to extend the previous characterization of the mean curvature motion to the non-radial case, namely to find a distance on (n − 1)-dimensional in-tegral varifolds for which the curves of maximal slope of the area functional (namely the total mass of the varifold) are Brakke’s flows. Actually this should also select non-pathological flows, since the notion of metric derivative is quite stronger than Brakke’s notion of generalized velocity and this should prevent pathological behaviours like instant disappearance of some sets.
However, finding a good distance in the general case seems to be really challenging. The following remark points out some problems in this direction.
Remark 2.0.6. In recent years many problems about gradient flows in metric spaces
have been studied using formal riemannian structures. For example, thanks to the Benamou-Brenier formula for the Wasserstein distance, one can view the metric space of probability measures with finite second moment P2(Rn) as an infinite dimensional
riemannian manifold whose tangent space at a point µ is given by vector-fields in L2µ (see Chapter 8 of [2]).
Then, the Wasserstein distance W2turns out to be the riemannian distance associated
to the L2µ riemannian metric on the vector-fields in the tangent space.
We can notice that our distance can be characterized in a similar way. In fact, it turns out that
dn((ri), (si))2 = inf ( 4ωn−1 (n + 1)2 +∞ X i=1 ˆ 1 0 vi(t) n+1 2 0 2 dt : vi(0) = ri, vi(1) = si ) = inf (+∞ X i=1 ˆ 1 0 |vi0(t)|2ωn−1vi(t)n−1dt : (vi(0)) = (ri), (vi(1)) = (si) ) = inf (+∞ X i=1 ˆ 1 0 |vi0(t)|2d(vi(t)) : (vi(0)) = (ri), (vi(1)) = (si) ) .
In view of the last formula, we can say that dn is the riemannian distance generated
by the L2(v
i)metric on a tangent space that is made of normal vector-fields on the support
of (vi) (here we are thinking at (vi) as a measure/varifold supported on the spheres with radii equal to (vi)).
This definition can be extended to the non-radial case, but unfortunately it turns out that this is not a good choice. Indeed, it was noticed by Michor and Mumford in [22] that this “distance” vanishes for every pair of embedded curves in R2. However, the paths
along which we have to compute the length to find a small value of the distance pass through some curves with very large area (that in this case is the length of the curve) that should not be admissible when one tries to decrease the area. So, a possibility could be to find some penalization that forbids these short but undesired paths without changing the infinitesimal behaviour of the distance (namely the “riemannian metric”).
3
The Modica-Mortola functionals
3.1
Γ-convergence
The Modica-Mortola functionals are defined, for every ε > 0, as
M Mε(u) := ˆ Rn ε|∇u(x)|2+1 εW (u(x)) dx,
where W is a so-called double-well potential. The model example is W (u) = (1 − u2)2, but we can also consider slightly more general potentials. More precisely, in this section we make the following assumptions on W :
• W ∈ C1(R),
• W (u) ≥ 0 for every u ∈ R and W (u) = 0 if and only if u = ±1, • lim inf
|u|→+∞W (u) > 0.
The following result was first proved in [23] for double-well potentials, relying on ideas from [24], where a different kind of potentials W (depending also on ε) was considered.
Theorem 3.1.1. It turns out that
Γ − lim
ε→0M Mε(u) =
(
σ · |∇u|(Rn)/2 if u ∈ BV (Rn, {±1}),
+∞ if u ∈ L2loc(Rn) \ BV (Rn, {±1}), where the Γ-limit is intended with respect to the L2loc topology and
σ := ˆ 1 −1 2 q W (u) du = ˆ +∞ −∞ ˙ q(s)2+ W (q(s))ds,
where q : R → [−1, 1] is a solution of the following equation
˙
q(s) =qW (q(s)), q(±∞) = ±1.
Proof. Liminf inequality: let {uε} ⊂ L2loc(Rn) be a family of functions and let u ∈
L2loc(Rn) be such that uε→ u in the topology of L2
loc. We have to prove that
lim inf
ε→0 M Mε(uε) ≥
(
σ · |∇u|(Rn)/2 if u ∈ BV (Rn, {±1}),
Up to restrict to a subsequence, we can assume that M Mε(uε) is bounded by a
constant M , otherwise the conclusion is trivial. Then the boundedness of the second addendum in the Modica-Mortola functionals implies that u(x) = ±1 for almost every
x ∈ Rn.
Now let us fix L > max{W (u) : u ∈ [−1, 1]} and let us set WL(u) := W (u) ∧ L and
GL(s) := ˆ s u0 2 q WL(u) du,
where u0is chosen in such a way that GL(±1) = ±σ/2 (u0exists because of the definition
of σ).
Then, by Young’s inequality and the chain rule we have that
M ≥ lim inf ε→0 ˆ Rn ε|∇uε|2+ 1 εW (uε) dx ≥ lim inf ε→0 ˆ Rn 2 q WL(uε)|∇uε| dx = lim inf ε→0 ˆ Rn |∇(GL◦ uε)| dx.
Since GL is Lipschitz continuous, then GL◦ uε→ GL◦ u in L2loc. Therefore, by the
lower semicontinuity of the BV-norm we have that
lim inf
ε→0 M Mε(uε) ≥ lim infε→0
ˆ
Rn
|∇(GL◦ uε)| dx ≥ |∇(GL◦ u)|(Rn).
Since u takes values in {±1}, then GL◦ u = σu/2 and hence
lim inf
ε→0 M Mε(uε) ≥ σ
|∇u|(Rn)
2 .
Limsup inequality: it is enough to prove the limsup inequality for the functions
u ∈ BV (Rn, {±1}) such that there exists a set E ⊆ Rnwith smooth boundary such that
u(x) =
(
1 if x ∈ E, −1 if x /∈ E,
because functions with this property are dense in energy in BV (Rn, {±1}) with respect
to the limit functional.
Let qnbe a sequence of smooth functions such that qn→ q in the sense of C1(R) and
qn(x) = q(x) ∀x ∈ [−n + 1, n − 1],
qn(x) = 1 ∀x ≥ n,
qn(x) = −1 ∀x ≤ −n.
Let us fix n ∈ N and let us set uε(x) := qn(dE(x)/ε), where dE is the signed distance
function from ∂E. Clearly uε→ u in L2
3.2. Metric slope
distance function from the smooth set ∂E and the coarea formula, for ε sufficiently small we have that M Mε(uε) = ˆ {x:|dE(x)|<εn} " qn0 d E ε 2 |∇d E|2 ε + 1 εW qn d E ε # dx = ˆ εn −εn 1 ε " qn0 t ε 2 + W qn t ε # Hn−1({x : dE(x) = t}) dt = ˆ n −n h qn0(s)2+ W (qn(s)) i Hn−1({x : dE(x) = εs}) ds.
Therefore it turns out that
lim sup ε→0 M Mε(uε) = Hn−1(∂E) ˆ n −n h qn0(s)2+ W (qn(s)) i ds.
Letting n → +∞ we deduce that
Γ − lim sup
ε→0
M Mε(u) ≤ σHn−1(∂E) = σ|∇u|(R n)
2 .
3.2
Metric slope
The aim of this section is to provide a rigorous computation of the metric slope of the Modica-Mortola functionals in a suitable metric space.
To this end we need more assumptions on the potential W than what we have used in the proof of the Γ-convergence result, so we also assume that:
• W0(u) is positive and nondecreasing for u ∈ (1, +∞),
• W0(u) is negative and nondecreasing for u ∈ (−∞, −1),
• W is λ-convex, namely there exists λ ∈ R such that W (u) + λu2 is convex,
• There exist two constants a, b ∈ R such that W0(u) ≤ aW (u) + b.
We point out that the third condition follows from the first one and the second one if W ∈ C2(R).
Moreover, we have to choose the correct metric space. This is not completely trivial, since we would like to use the L2 distance, but the functions we are dealing with do not belong to L2(Rn), because they are asymptotically equal to ±1 almost everywhere.
Therefore, it would be natural to work in the space L2loc(Rn), as we did for the Γ-convergence, but the usual distances that metrize this space are not nice for our purposes. So, we use L2loc(Rn) as functional space but we endow it with a truncation of the L2 distance, namely we set
d(u, v) := minnku − vkL2(Rn), 1
o
We point out that the notion of convergence (and the topology) associated to this distance is stronger than the usual one of L2loc(Rn), since we clearly have that
un→ u ⇐⇒ kun− ukL2(Rn)→ 0.
At this point we are almost ready to compute the metric slope of the Modica-Mortola functionals, but we first need the following regularity result for solutions of a semilinear PDE.
Lemma 3.2.1. Let R > 0 be a positive radius and let f ∈ L2(BR). Let u ∈ H1(BR) be
a weak solution of the following PDE
−∆u + 1 2W
0(u) = f.
Let us assume also that ˆ
BR
W (u(x)) dx < +∞.
Then it turns out that W0(u) ∈ L2loc(BR) and therefore u ∈ Hloc2 (BR).
Proof. Let us fix r < R and let ϑ ∈ Cc∞(BR) be a smooth cutoff function for Br, namely such that ϑ(x) = 1 for every x ∈ Br and 0 ≤ ϑ(x) ≤ 1 for every x ∈ BR.
Let us also define the following sequence of functions
gn(u) := W (u)2n+11 W0(u)1− 1 2n if u ≥ 1, 0 if u ≤ 1.
We point out that the assumptions on W imply that all the functions gn are
non-decreasing.
We now claim that for every n ∈ N it holds that (gn◦ u)ϑ ∈ L2(BR) and, if n ≥ 1,
then ˆ BR gn(u(x))2ϑ(x)2dx ≤ C ˆ BR gn−1(u)2ϑ(x)2dx !12 , (3.2.2) where C = Ckf kL2(B R), k∇ϑkL∞(BR), kukH1(BR)
is a positive constant that does not depend on n.
We now prove this claim by induction. The base case n = 0 follows directly from the assumption on the integrability of W (u).
As for the inductive step, let us assume that (3.2.2) holds for some (fixed) n ∈ N. Let gn,m : R → [0, +∞) be a sequence nondecreasing, smooth and bounded functions such that gn,m % gn, as m → +∞.
Since gn,m is smooth and ϑ vanishes on ∂BR, we have that (gn,m◦ u)ϑ2 ∈ H01(BR).
Therefore we can use (gn,m◦ u)ϑ2 as a test function for the PDE, and hence we deduce
that ˆ BR f gn,m(u)ϑ2= ˆ BR (∇u · ∇((gn,m◦ u)ϑ2)) + W0(u) 2 gn,m(u)ϑ 2 = ˆ BR g0n,m(u)ϑ2|∇u|2+ 2ϑg n,m(u)(∇u · ∇ϑ) + W0(u) 2 gn,m(u)ϑ 2.
3.2. Metric slope
Therefore, since gn,m0 is non-negative we have that ˆ BR W0(u)gn,m(u)ϑ2≤ 2 ˆ BR f gn,m(u)ϑ2− 2ϑgn,m(u)(∇u · ∇ϑ) ≤ 2kf ϑkL2(B R)+ 2k(∇u · ∇ϑ)kL2(BR) ˆ BR gn,m(u)2ϑ2 !12 ≤ 2kf kL2(B R)+ 2kukH1(BR)k∇ϑkL∞(BR) ˆ BR gn,m(u)2ϑ2 !12 .
By the monotone convergence theorem, taking the supremum over all m ∈ N we deduce that ˆ BR W0(u(x))gn(u(x))ϑ(x)2dx ≤ C ˆ BR gn(u(x))2ϑ(x)2dx !12 .
By definition of gn+1 this means exactly that ˆ BR gn+1(u(x))2ϑ(x)2dx ≤ C ˆ BR gn(u(x))2ϑ(x)2dx !12 ,
and this concludes the proof of (3.2.2) by induction.
We can now apply n times inequality (3.2.2) to deduce that for every n ∈ N we have that ˆ BR gn(u(x))2ϑ(x)2dx ≤ C2− 1 2n−1 ˆ BR g0(u)2ϑ(x)2dx !2n1 .
Since we can clearly assume that C ≥ 1, then for every n ∈ N we also deduce that ˆ BR gn(u(x))2ϑ(x)2dx ≤ C2 ˆ BR g0(u)2ϑ(x)2dx !1 2n .
Recalling the definition of gn and the properties of ϑ we have proved that
ˆ Br W (u(x))2n1 W0(u(x))2− 1 2n−11{u≥1}(u(x)) dx ≤ C2 ˆ BR W (u(x)) dx !2n1 .
Therefore, since the right-hand side is eventually bounded by 2C2 and, by the
prop-erties of W , we have that W (u(x)) ≥ W (2) > 0 where u(x) ≥ 2, we can deduce that if
n is large then
ˆ
Br
W0(u(x))2−2n−11 1{u≥2}(u(x)) dx ≤
2C2 W (2)2n1
≤ 4C2.
Hence we can apply the monotone convergence theorem to pass to the limit the integral on the set {x ∈ Br: W0(u(x)) > 1} and the dominated convergence theorem to
pass to the limit the integral on the set {x ∈ Br: W0(u(x)) ≤ 1} to deduce that ˆ
Br
and this means that W0(u)1{u≥2} ∈ L2(Br). Now, if we define ˜ gn(u) := W (u)2n+11 (−W0(u))1− 1 2n if u ≤ −1, 0 if u ≥ −1,
then an analogous argument leads to −W0(u)1{u(x)≤−2}∈ L2(Br) and therefore we have
that W0(u) ∈ L2(Br). Since r can be any radius less than R, then we have proved that
W0(u) ∈ L2
loc(BR), and therefore −∆u ∈ L2loc(BR). By basic elliptic regularity theory
this implies also that u ∈ Hloc2 (BR).
We can now compute the metric slope of the Modica-Mortola functionals.
Proposition 3.2.2. Let us fix ε > 0 and let u ∈ Hloc1 (Rn) be a function such that
M Mε(u) is finite. Then it turns out that
|∇M Mε|(u) = r´ Rn −2ε∆u(x) +1 εW0(u(x)) 2 dx if u ∈ Hloc2 (Rn), +∞ otherwise,
where the slope is computed with respect to the distance defined in (3.2.1).
Proof. Let us assume that |∇M Mε|(u) = M < +∞. This means that for every smooth
function ϕ ∈ Cc1(Rn) we have that
M = lim sup v→u (M Mε(u) − M Mε(v))+ ku − vkL2(Rn) ≥ lim sup t→0 (M Mε(u) − M Mε(u + tϕ))+ ktϕkL2(Rn) . (3.2.3)
By an elementary computation we deduce that
M Mε(u) − M Mε(u + tϕ) = ˆ Rn −2tε(∇u · ∇ϕ) − t2ε|∇ϕ|2+W (u) − W (u + tϕ) ε dx.
Dividing both sides by |t| and letting t → 0 we deduce that
lim sup t→0 (M Mε(u) − M Mε(u + tϕ))+ |t| = ˆ Rn 2ε(∇u · ∇ϕ) + W 0(u)ϕ ε dx . (3.2.4)
We point out that we can pass to the limit in the term with W using the dominated convergence theorem. The required domination is guaranteed by the growth assumption on W0 and the fact that ϕ has compact support and is bounded.
Therefore, by (3.2.3) and (3.2.4) we deduce that for every function ϕ ∈ Cc1(Rn) we have that ˆ Rn 2ε(∇u · ∇ϕ) + W 0(u)ϕ ε dx ≤ M kϕkL2(Rn).
By Riesz’s theorem this implies that the distribution −2ε∆u + W0(u)/ε is actually a function in L2(Rn). By Lemma 3.2.1 (applied to a suitable rescaling of u that cancels the ε), then W0(u) ∈ L2loc(Rn) and u ∈ Hloc2 (Rn).
3.2. Metric slope
Moreover, the last inequality also implies that
k − 2ε∆u + W0(u)/εkL2(Rn)≤ M = |∇M Mε|(u).
In order to prove the other inequality, we can assume that u ∈ Hloc2 (Rn), otherwise the conclusion is trivial. In this case, by Gauss-Green formula we have that
M Mε(u) − M Mε(v) ≤ ˆ Rn −2ε(∇u · (∇v − ∇u)) + 1 ε(W (u) − W (v)) dx ≤ ˆ Rn −2ε∆u(u − v) +1 ε(W (u) − W (v)) dx.
Moreover, by the λ-convexity of W , we have that
W (u) − W (v) ≤ W0(u)(u − v) + λ(u − v)2,
and hence M Mε(u) − M Mε(v) ≤ ˆ Rn −2ε∆u(u − v) +1 εW 0(u)(u − v) +λ ε(u − v) 2 dx.
Therefore we can estimate the metric slope from above in the following way
|∇M Mε|(u) = lim sup
v→u (M Mε(u) − M Mε(v))+ ku − vkL2(Rn) ≤ lim sup v→u ´ Rn −2ε∆u(u − v) + 1 εW 0(u)(u − v) dx + λ εku − vk2L2(Rn) ku − vkL2(Rn) = lim sup v→u ´ Rn −2ε∆u + 1εW0(u)(u − v) dx ku − vkL2(Rn) +λ εku − vkL2(Rn) ≤ lim sup v→u −2ε∆u + 1 εW 0(u) L2(Rn)ku − vkL2(Rn) ku − vkL2(Rn) = −2ε∆u + W0(u)/ε L2(Rn),
and this concludes the proof.
3.2.1 Metric slope in radial symmetry
Now we want to show that in the case of radial symmetry the metric slope of the Modica-Mortola functionals can be written in a particular way that is very useful to estimate its Γ-liminf.
Definition 3.2.3. We say that u : Rn→ R is a radial function if there exists a function ˜
u : [0, +∞) → R such that u(x) = ˜u(|x|) for every x ∈ Rn.
In the sequel, we will denote ˜u and u with the same letter u. So, if r ∈ [0, +∞) is a
Lemma 3.2.4. Let us fix ε > 0 and let u ∈ Hloc2 (Rn) be a radial function. Then it turns
out that
lim
r→0+u(r)˙
2rn−2= 0.
Proof. Let us set f := ∆u. We know that f ∈ L2loc(Rn). Moreover, we have that ¨
u(r) + n − 1
r u(r) = f (r).˙
Multiplying both sides by rn−1 we find that ( ˙urn−1)0 = f (r)rr−1,
from which we can deduce that there exists a constant c ∈ R such that the following formula for ˙u holds
˙ u(r) = 1 rn−1 c + ˆ r 0 f (s)sn−1ds . (3.2.5)
We now prove that c = 0. To this end, we use the fact that ˙u ∈ L2loc(Rn) and some elementary computations to deduce that the following integrals are all finite
ˆ 1 0 ˙ u(r)2rn−1dr = ˆ 1 0 1 rn−1 c + ˆ r 0 f (s)sn−1ds 2 dr ≥ ˆ 1 0 c2 2rn−1 − 1 rn−1 ˆ r 0 f (s)sn−1ds 2 dr ≥ ˆ 1 0 c2 2rn−1 − 1 rn−1 ˆ r 0 f (s)2sn−1ds ˆ r 0 sn−1ds dr = ˆ 1 0 c2 2rn−1dr − ˆ 1 0 r n ˆ r 0 f (s)2sn−1ds dr.
Since the last integral is finite, while the integral involving c is divergent if c 6= 0, it must be c = 0.
Therefore, recalling (3.2.5), we can compute the limit of ˙u(r)2rn−1 using Cauchy-Schwarz inequality and the fact that f ∈ L2
loc(Rn), as follows lim r→0+u(r)˙ 2rn−2= lim r→0+ 1 rn ˆ r 0 f (s)sn−1ds 2 ≤ lim r→0+ 1 rn ˆ r 0 sn−1ds ˆ r 0 f (s)2sn−1ds = lim r→0+ 1 n kf k2 L2(B r) ωn−1 = 0.
Corollary 3.2.5 (Slope for n = 2). Let us assume that the space dimension is n = 2.
Let us fix ε > 0 and let u ∈ L2loc(R2) be a radial function such that both M Mε(u) and
|∇M Mε|(u) are finite. Then it turns out that 1 ε|∇M Mε|(u) 2 = ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε 2 2πr dr + ˆ +∞ 0 4ε ˙u2 r 2π dr + 8πW (u(0)) ε .
3.2. Metric slope
Proof. By Proposition 3.2.2 we know that u ∈ Hloc2 (R2) and, using polar coordinates to compute the integral in the formula for the slope, we obtain that
1 ε|∇M Mε|(u) 2= ˆ +∞ 0 2√ε¨u − W 0(u) ε√ε + 2√ε r u˙ !2 2πr dr = ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε 2 2πr dr + ˆ +∞ 0 4ε ˙u2 r 2π dr + ˆ +∞ 0 2 2√ε¨u −W 0(u) ε√ε 2√ε ˙u 2π dr.
Since the function in the last integral is actually a derivative, we have that ˆ +∞ 0 2 2√ε¨u −W 0(u) ε√ε 2√ε ˙u 2π dr = 8π ε ˙u(r)2−W (u(r)) ε r=+∞ r=0 .
By the finiteness of M Mε(u) we can easily deduce that the term with r = +∞
vanishes. Moreover Lemma 3.2.4 implies that ˙u(0) = 0. Therefore we have that
ˆ +∞ 0 2 2√ε¨u − W 0(u) ε√ε 2√ε ˙u 2π dr = 8πW (u(0)) ε ,
and this concludes the proof.
Corollary 3.2.6 (Slope for n ≥ 3). Let us assume that the space dimension is n ≥ 3.
Let us fix ε > 0 and let u ∈ L2loc(R2) be a radial function such that both M Mε(u) and
|∇M Mε|(u) are finite. Then it turns out that 1 ε|∇M Mε|(u) 2 ≥ ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε 2 ωn−1rn−1dr+ ˆ +∞ 0 4(n − 1)ε ˙u2+ 4(n − 1)(n − 2)W (u) ε ωn−1rn−3dr.
Proof. By Proposition 3.2.2 we know that u ∈ Hloc2 (R2) and, using polar coordinates to compute the integral in the formula for the slope, we obtain that
1 ε|∇M Mε|(u) 2 = ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε + 2√ε(n − 1) r u˙ !2 ωn−1rn−1dr = ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε 2 ωn−1rn−1dr + ˆ +∞ 0 4ε(n − 1)2u˙2ωn−1rn−3dr + ˆ +∞ 0 2 2√ε¨u −W 0(u) ε√ε 2√ε(n − 1) ˙u ωn−1rn−2dr.
Integrating the last term by parts we deduce that ˆ +∞ 0 2 2√ε¨u − W 0(u) ε√ε 2√ε(n − 1) ˙u ωn−1rn−2dr = 4 ε ˙u(r)2− W (u(r)) ε (n − 1)ωn−1rn−2 r=+∞ r=0 − ˆ +∞ 0 4 ε ˙u2−W (u) ε (n − 1)ωn−1(n − 2)rn−3dr.
By the finiteness of M Mε(u) we can easily deduce that the term with r = +∞
vanishes. Moreover, Lemma 3.2.4 implies that also the term with ˙u(r)rn−2 vanishes in
r = 0. Since the term with W (u(r))rn−2at r = 0 has positive sign, then the contribution of the boundary term is positive. Therefore we have that
1 ε|∇M Mε|(u) 2 = ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε + 2√ε(n − 1) r u˙ !2 ωn−1rn−1dr ≥ ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε 2 ωn−1rn−1dr + ˆ +∞ 0 4ε(n − 1)2u˙2ωn−1rn−3dr − ˆ +∞ 0 4 ε ˙u2−W (u) ε (n − 1)ωn−1(n − 2)rn−3dr = ˆ +∞ 0 2√ε¨u −W 0(u) ε√ε 2 ωn−1rn−1dr+ ˆ +∞ 0 4(n − 1)ε ˙u2+ 4(n − 1)(n − 2)W (u) ε ωn−1rn−3dr.
Remark 3.2.7. For every space dimension n ≥ 2, we have that
1 ε|∇M Mε|(u) 2 ≥ ˆ +∞ 0 2√ε¨u(r) − W 0(u(r)) ε√ε 2 ωn−1rn−1dr + ˆ +∞ 0 4(n − 1)ε ˙u(r)2ωn−1rn−3dr.
4
Γ-liminf of metric slopes
The aim of this chapter is to provide an estimate from below for the Γ-liminf of the metric slope of the Modica-Mortola functionals and a characterization of the asymptotic behaviour of the families uε for which this Γ-liminf is finite. This is one of the main steps in order to study convergence of the solutions of the Allen-Cahn equation.
The main idea to prove such an estimate is to look at the behaviour of uε near its
zeros. To this end, we study the behaviour of a suitable rescaling of uε that is usually called blow-up. The precise definition is the following.
Definition 4.0.1 (Blow-up). Let uε : Rn → R be a family of radial functions and let
{Rε} ⊂ [0, +∞) be a family of non-negative real numbers. Then the blow-up of uε at
Rε is the family of functions ψε: [−Rε/ε, +∞) → R defined as
ψε(s) := uε(Rε+ εs).
The first lemma that we prove characterizes the solutions of the ordinary differential equation corresponding to the vanishing of the slope in the one-dimensional case. This is also the equation that the blow-up should verify in the limit for ε → 0.
In the sequel, we reduce to the case in which W (u) = (1 − u2)2, since this makes many statements and proofs much clearer, because in this case we can solve explicitly the one-dimensional equation. However, with proper modifications, all the results hold also for more general double-well potentials W .
Lemma 4.0.2. Let ψ : R → R be a globally defined solution of the following ordinary
differential equation
2 ¨ψ = W0(ψ) = −4ψ(1 − ψ2). (4.0.1)
Let us assume in addition that
ˆ +∞ −∞
˙
ψ(s)2+ W (ψ(s))ds < +∞.
Then one of the following holds
• ψ(s) ≡ +1,
• ψ(s) ≡ −1,
• there exists c ∈ R such that ψ(s) = tanh(s − c), • there exists c ∈ R such that ψ(s) = − tanh(s − c).
