DIPARTIMENTO DI MATEMATICA Corso di Laurea in Matematica
Tesi di laurea magistrale
Gradient flow of the one-dimensional Mumford-Shah
functional as a Curve of Maximal Slope
Candidata
Clara Antonucci
Relatore
1 Introduction 1
1.1 Possible approaches to the Gradient Flow of the
Mumford-Shah functional . . . 1
1.1.1 The Maximal Slope Curve method . . . 1
1.1.2 The minimizing movements method . . . 2
1.2 Previous literature . . . 3
1.3 General theory on the Mumford-Shah functional . . . 6
1.4 The discrete setting . . . 9
1.5 General results . . . 10
2 General theory of Gradient Flows in metric spaces 13 2.1 Definition of descending metric slope and main properties . . . 13
2.2 Absolutely Continuous curves in metric spaces . . . 14
2.3 Curves of Maximal Slope in metric spaces . . . 22
3 Variational approximations of the Mumford-Shah functional 43 3.1 A class of suitable functions . . . 43
3.2 Gamma-convergence . . . 46
3.2.1 Limsup inequality . . . 46
3.2.2 Liminf inequality . . . 47
3.3 Equicoerciveness . . . 58
4 Convergence of metric slopes 63 4.1 Slope of the approximating functionals . . . 63
4.2 Slope of the Mumford-Shah functional . . . 63
4.3 Convergence of metric slopes . . . 64
4.4 Functions with jumps with multiplicity . . . 76
5 Gradient Flow of the Mumford-Shah functional 79 5.1 Gradient Flow of the Dirichlet functional . . . 79
5.2 Gradient Flow of the Mumford-Shah functional . . . 84
5.2.1 A general lower semicontinuity result . . . 84
5.2.2 Uniqueness of solution before the first jump “disappears” . . . 86
5.2.3 Lack of uniqueness after the first jump “disappears” and possible behaviours . . . 94
6 Convergence of Gradient Flows 97 6.1 Estimates for fixed n . . . . 98
6.2 Definition and first properties of u∞(t) . . . 103
6.3 Convergence of Gradient Flows . . . 105
1
Introduction
1.1
Possible approaches to the Gradient Flow of the
Mumford-Shah functional
The goal of this thesis is to approach the problem of the Gradient Flow of the one-dimensional Mumford-Shah functional as a Curve of Maximal Slope. We recall that the Mumford-Shah functional on an open set Ω ⊆ Rdis defined as follows
MS(u) := ˆ
Ω
|∇u (x)|2 dx + Hd−1(Su) ,
where ∇u (x) denotes the approximate gradient of u, Su denotes the set of the essential
discontinuities of u and Hd−1 denotes the (d − 1)-dimensional Hausdorff measure (we refer to Section 1.3 for more details). This functional was introduced by D. Mumford and J. Shah in 1989 in [21].
A quite natural setting for Gradient Flow problems is a Hilbert space H and a C1 function F . We fix an initial datum x0 ∈ H and we are looking for solutions of the
system
( d
dtu(t) = −∇F (u(t)), u(0) = x0.
In recent years more and more interest has been focused on a generalization of this problem, in order to weaken the requests both on the ambient space and on the regularity of F .
1.1.1 The Maximal Slope Curve method
In a generic metric space (X, d) there are no vectors or directions, just points. It makes no sense to consider the gradient of a function F : X → R as an element of X, but what is easy to generalize is the equivalent of the norm of the gradient, and this has led to the definition of the descending metric slope, defined by E. De Giorgi, A. Marino and M. Tosques in [15] as follows
|∇F |(x0) := lim sup
r→0+
F (x0) − inf {F (x) : d(x, x0) ≤ r}
r .
Intuitively, it is clear that one can mimic the Gradient Flow system in a generic metric space by choosing the curve along which the function F decreases as fast as possible.
This leads to the concept of Curve of Maximal Slope. It is easy to prove that in a Hilbert space the following implication holds
(
u(t) and F are of class C1,
− d dtF (u(t)) ≥ 1 2k∇F (u(t))k 2 H+ 12k ˙u(t)k 2 H, =⇒ (
−dtdF (u(t)) = 12k∇F (u(t))k2H+ 12k ˙u(t)k2H,
∇F (u(t)) = − ˙u(t).
(for a complete discussion of this, we refer to Chapter 2, Section 2.3).
This differential request can be translated into an integral request asking that for every 0 ≤ t1 < t2 it holds that
F (u(t1)) − F (u(t2)) ≥ 1 2 ˆ t2 t1 k∇F (u(t))k2H dt + 1 2 ˆ t2 t1 k ˙u(t)k2H dt. (1.1.1)
Now we notice that the request of (1.1.1), that in the regular case is equivalent to the Gradient Flow equation, also makes sense in the very weak context in which (X, d) is just a metric space and F is any function. In fact we can replace the norm of the gradient with the descending metric slope and the norm of the derivative with the metric derivative. We refer to Chapter 2, Section 2.3 for the development of this strategy.
1.1.2 The minimizing movements method
Another possible way to overcome the lack of an affine structure is the so-called mini-mizing movements method, introduced by E. De Giorgi in [14].
The basic idea of this strategy is to consider a discretization in time (for example of step n1) and to define an approximate solution inductively, asking that
un(0) = x0, un k + 1 n ∈ argmin F (x) + n 2d x, un k n 2! . (1.1.2)
Under very mild hypotheses, if problem (1.1.2) is set in a Hilbert space with a C1 function F , then the limit curve as n approaches infinity is a solution of the Gradient Flow system. In fact, if we can compute the derivative of (1.1.2) and if we assume that the point x is a minimizer, then we get that
0 = ∇ " F (x) + n 2d x, u k n 2# |x=x = ∇F (x) + n x − u k n ,
that is to say that uk+1n − ukn 1 n = −∇F u k + 1 n .
The previous equation is the Euler backward approximation of step n1 of the Gradient Flow equation
d
dt(u(t)) = −∇F (u(t)),
therefore it is reasonable to expect that, under general assumptions, it converges to a solution of the Gradient Flow equation.
Now we notice that the discrete system (1.1.2) also makes sense in the very weak context in which (X, d) is just a metric space and F is a lower semicontinuous and locally coercive function.
Therefore one can solve problem (1.1.2), show that the limit is an (absolutely) con-tinuous curve and define the Gradient Flow of the function F in the metric space (X, d) as any possible limit of this process. It is also possible to show that these possible limits are Curves of Maximal Slope for the function F (see Chapter 2 for a precise proof).
1.2
Previous literature
In the case of convex functionals (as well as functionals that satisfy some suitable gener-alizations of convexity) there are some relatively simple strategies that allow us to define the Gradient Flow, and also the numerical computation of the minimizers is simpler. We do not want to provide more details about the general theory for convex functionals here, the interested readers can find more in [2].
This is the reason why the Gradient Flow of the Dirichlet functional with respect to the L2 distance has been fully understood. Indeed, it is well known that the Gradient Flow of the Dirichlet functional with respect to the L2 distance (at least for smooth do-mains) coincides with the unique solution of the heat equation with Neumann Boundary Conditions.
Intuitively, thanks to the obvious connections between the Dirichlet functional and the Mumford-Shah functional (which is not convex), one might expect the same be-haviour for the Gradient Flow of the Mumford-Shah functional, i.e. that the following equation holds
∂tu(t, x) = 2∆xu(t, x),
at least in a weak sense and for points x that do not belong to the set of discontinuities for some open interval of time. One also might expect that the Neumann Boundary Conditions hold
∂nu(t, x) = 0
both on the boundary of Ω and on the set of discontinuities of u. However, at the present time, this remains just a mathematical intuition, and only some particular cases have been investigated fully.
Now we would like to mention some of the main contributions that have appeared in the literature in the last decades. First of all, however, it is convenient to discuss
an important matter that divides the possible definitions of the Gradient Flow of the Mumford-Shah functional into two categories, and this concerns the assumtpion of
irre-versibility.
The Mumford-Shah energy can be seen as an elastic energy in a model of elasto-brittle bodies. In this context, Ω represents a body of some material that is elastic up to a maximum threshold. If the body undergoes an effort that goes beyond its physical maximum elasticity, then it breaks, creating a fracture (that we assume in general to be a decent set of Hausdorff dimension d − 1). The energy that we have used to perform the fracture is not accumulated in the body as elastic energy, but it is not lost. Indeed, we “remember” this energy by considering the (d − 1)-dimensional volume of the crack. Therefore, from a physical point of view, it makes sense to consider the unilateral model, hence when the body develops a fracture we assume that that portion of the body stays broken for all future times.
In this context, the evolution goes on separately on one side of the fracture and the other, and even if the two edges happen to be perfectly aligned and contiguous to each other, nevertheless the crack cannot be fixed and remains forever.
These ideas have their roots in the theory of fracture mechanics, which was introduced by A. A. Griffith in 1920 (see [18]).
Another approach to the Gradient Flow of the Mumford-Shah functional is however also possible. From the pure mathematical point of view, we can allow the set of discon-tinuities of the function u to vary without constraints. It is therefore a priori possible that
• the fracture moves, and (in dimension strictly greater than one) becomes larger or smaller,
• some portion of the fracture instantly disappears (this is easy to picture in dimen-sion one, since the fractures are just points).
This model is the most natural mathematical translation of the Gradient Flow of the Mumford-Shah functional, since no additional requests are made on the functional, and this is the path that we follow in this thesis.
Now we can finally present a brief list of the most important papers that have dealt with the Gradient Flow of the Mumford-Shah functional.
In 1997 A. Chambolle and F. Doveri studied the problem in dimension two in the irreversible context (see [12]). They proved that there is a unique possible limit of the minimizing movements and that it solves the heat equation except in a set whose Hausdorff dimension is controlled from above (under some hypotheses). They also proved an inequality in the spirit of Maximal Slope Curves.
In 1998 M. Gobbino (see [17]) studied a family of approximating functionals (pro-posed some years before by E. De Giorgi) that Gamma-converges to the one-dimensional Mumford-Shah functional (see Chapter 3 for more details). He then studied the Gradient Flows of these regular discrete approximations, defining the Gradient Flow of the one-dimensional Mumford-Shah functional as any possible limit of these flows. He found a
sufficient condition for the uniqueness of the possible limits and explained some patholo-gies that arise in the case in which uniqueness does not hold. For a deeper description of this strategy, see Chapter 6.
In 2002 M. Morini studied the problem in any space dimension using the strategy of
calibrations (see [19]). Morini did not work under assumtpions of irreversibility, however
he proved that, if the initial datum is sufficiently regular (both for the discontinuity set and for the regularity outside), then the set of discontinuities does not change for small times. He proved that the solution solves the heat equation outside this set. This proof has the advantage to work in every dimension, but it is surely not the best in the one-dimensional case, in which much simpler proofs are possible.
In 2003 G. Alberti, G. Bouchitt´e and G. Dal Maso (see [1]) studied a sufficient condition for the minimality of a general function u, also working with calibrations. They proved that, if it is possible to find a vectorfield that satisfies suitable hypotheses, then the function u is a minimizer; however, these conditions are in general not so easy to check. Under some regularity assumptions (that imply continuity in the whole Ω) on the initial datum, they proved that there is a unique possible limit of the minimizing movement systems, and it solves the heat equation without developing discontinuities.
In 2006 G. Dal Maso and R. Toader presented a paper that is of great relevance in this context, although it does not involve directly the Gradient Flow of the Mumford-Shah functional. Indeed, they studied the unilateral slope for the Mumford-Shah functional in every space dimension d. The unilateral slope for the Mumford-Shah functional computed in some function u coincides with the usual descending metric slope, with the only difference that all the competitors are forced to pay not only for their Dirichlet energy and for the (d − 1)-dimensional volume of their discontinuity set, but also for the (d − 1)-dimensional volume of the discontinuity set of u. In this way, it is clear that the competitors to the infimum “try to have the same discontinuity set” of the function u. For more details, see [13].
In 2014 J. F. Babadjian and V. Millot (see [5]) followed the unilateral model, working only in dimension two and studying the approximation introduced by L. Ambrosio and V. M. Tortorelli in [3] and the limit of their minimizing movements. Asking that the set of discontinuities at the initial time is compact and connected, they obtained that the limit function is a Maximal Slope Curve.
In 2018 A. Braides and V. Vallocchia (see [8]) studied the Gradient Flow of the one-dimensional Mumford-Shah functional. They used a result of M. Morini and M. Negri (see [20]) in which the authors showed that a suitable rescaled discretization of the Perona-Malik functionals Gamma-converges to the Mumford-Shah functional in di-mension one and two. Therefore they studied the Gradient Flow of these approximating sequences in dimension one and investigated some hypotheses under which these flows converge to the Gradient Flow of the one-dimensional Mumford-Shah functional. Their
dissipation principle is the discrete analogue of the irreversibility request, so they work
under the increasing fracture type assumption. The result of convergence of flows that they obtained is proved only until the first collision time, i.e. the first time for which the number of jumps of the limit function decreases. A full understanding of what can happen to the convergence of flows after the first collision time seems to be a challenging question and we refer to Chapter 6 for some perspectives.
Finally, we want to point out that there exists a huge literature concerning the numerical aspects of the Mumford-Shah functional, both for the computations of mini-mizers and for the study of Gradient Flow. The most interesting approximation of the Mumford-Shah functional from the numerical point of view seems to be the one of L. Am-brosio and V. M. Tortorelli. Since this thesis does not involve numerical considerations, we prefer not to go through these aspects.
The connections with the two-dimensional Mumford-Shah functional and image pro-cessing are also of great relevance and have been deeply investigated in existing literature, also numerically. For a brief description of the Mumford-Shah functional and its connec-tions with image reconstruction and edge detection (that are indeed the physical models that motivated the introduction of this functional) we refer to the original article [21].
1.3
General theory on the Mumford-Shah functional
The topic of this thesis is the one-dimensional Mumford-Shah functional, of which we now recall the simplest definition.
Definition 1.3.1 (One-dimensional Mumford-Shah functional). The Mumford-Shah
functional on an interval I ⊆ R is a functional
MS : L2(I) → [0, +∞] .
If u ∈ L2(I), then MS(u) is finite if and only if the set of essential discontinuities of u (denoted from now on with Su) is finite and the restriction of u to the complement set
is of class H1. In this case
MS(u) := ˆ
I\Su
u0(x)2 dx + |Su| .
Otherwise MS(u) := +∞.
We prefer to denote the weak derivative of the function u in [0, 1] with the symbol
u0 instead of ˙u, because in the following chapters we consider curves u(t, x), and the
symbol ˙u resembles a time derivative.
In the sequel we often refer to the essential discontinuities of u as “jumps”, since the above definition implies that in every point of essential discontinuity x0 these two limits
exist and are different
u(x−0) := lim x→x−0 u(x), u(x+0) := lim x→x+0 u(x).
This is due to the fact that in one-dimensional domains the H1 regularity implies a C0,12
regularity.
However, this definition is just the one-dimensional translation of a more general definition, which we now present for the sake of completeness.
Let d be any natural number and Ω ⊆ Rdbe an open set. For the sake of simplicity, we also assume that Ω is bounded, otherwise all the definitions and statements that we present must be interpreted locally.
Definition 1.3.2 (BV). A function u ∈ L1(Ω) belongs to BV (Ω) if there exists a measure µ ∈ MΩ, Rdsuch that
ˆ Ω div η(x)u(x) dx = − ˆ Ω η(x) dµ(x)
for every smooth vectorfield η ∈ Cc∞Ω, Rd. Such measure, if it exists, is unique. It is called distributional derivative of u and it is denoted by Du.
Now we define the set of Special Bounded Variations functions. For this definition, we closely follow [7].
Definition 1.3.3 (SBV). A function u ∈ BV (Ω) belongs to SBV (Ω) if its distributional
derivative can be written as
Du = µ + ν,
where µ is absolutely continuous with respect to the d-dimensional Lebesgue measure and ν can be written as
ν( dx) = g(x)Hd−1
|K( dx),
where K is a countable union of sets with finite (d − 1)-dimensional Hausdorff measure and g : Ω → Rn is of class L1 with respect to the measureHd−1
|K on Ω.
Definition 1.3.4 (Mumford-Shah functional). Let d be any natural number and Ω ⊆ Rd
be an open set. The Mumford-Shah functional on Ω is a functional MS : L2(Ω) → [0, +∞] defined as follows MS(u) := ( ´ Ω|∇u (x)| 2 dx + Hd−1(S u) ∈ [0, +∞] if u ∈ SBV (Ω) , +∞ if u /∈ SBV (Ω) ,
where ∇u (x) denotes the approximate gradient of u, Su denotes the set of the essential
discontinuities of u and Hd−1 denotes the (d − 1)-dimensional Hausdorff measure. Now we state a lower semicontinuity result that is well known.
Theorem 1.3.5 (Lower semicontinuity). The Mumford-Shah functional is lower
semi-continuous with respect to the L2 convergence, i.e. if un L
2
→ u∞, then
lim inf
n→+∞MS(un) ≥ MS(u∞).
From now on, we focus our attention on the one-dimensional Mumford-Shah functional (in particular we fix the domain to be the interval [0, 1]) and we present some general results that we use in the following chapters. In order to shorten the notations, it is useful to give the following definition.
Definition 1.3.6 (Piecewise H1 functions). We define P H1(0, 1) :=nu : [0, 1] → R s.t. u|(xi,xi+1) is H 1, for some 0 = x 0< · · · < xr= 1 o .
By definition, this set coincides with the set in which the one-dimensional Mumford-Shah functional is finite.
Proposition 1.3.7 (A density result). The set
P A(0, 1) :=nu : [0, 1] → R s.t. u|(xi,xi+1) is affine, for some 0 = x0< · · · < xr= 1
o
,
(1.3.1) is dense in energy for the one-dimensional Mumford-Shah functional, i.e. for every u ∈
L2(0, 1) there exists a sequence vn⊆ P A(0, 1) such that
• vn→ u with respect to the L2 convergence,
• MS(vn) → MS(u).
Lemma 1.3.8. Let us consider x0 ∈ [0, 1], y1 6= y2. Without loss of generality, we
assume that y1 < y2. We consider the following rectangles
R1 := [x0− δ1, x0] × [y1− α1, y1+ ε1] ,
R2 := [x0, x0+ δ2] × [y2− ε2, y2+ α2] .
defined for αi≥ 0, εi≥ 0, ε1+ ε2 < y2− y1.
Let us consider any u ∈ P H1(0, 1) and let us assume that
graph(u) ∩ R16= ∅, graph(u) ∩ R26= ∅, u is continuous in [x0− δ1, x0+ δ2] .
Then it holds that
MS(u|[x
0−δ1,x0+δ2]) ≥
(y2− y1− ε1− ε2)2
δ1+ δ2
. Proof. Let us consider
( (xP, yP) ∈ graph(u) ∩ R1, (xQ, yQ) ∈ graph(u) ∩ R2. It is clear that MS(u|[x0−δ1,x0+δ2]) ≥ MS(u|[x P,xQ]) ≥ MS(v),
where v is the affine function joining the points (xP, yP) and (xQ, yQ), by classical
convexity arguments (we point out that the Mumford-Shah functional computed on a continuous function coincides with the Dirichlet functional).
Therefore MS(u|[x 0−δ1,x0+δ2]) ≥ (yQ− yP)2 xQ− xP .
But we know that (xP, yP) ∈ R1 and (xQ, yQ) ∈ R2, therefore yQ− yP ≥ y2− y1− ε1− ε2, xQ− xP ≤ δ1+ δ2, so MS(u|[x0−δ1,x0+δ2]) ≥ (y2− y1− ε1− ε2)2 δ1+ δ2
and the proof is complete.
1.4
The discrete setting
In Chapter 3 we introduce a family of discrete approximations of the one-dimensional Mumford-Shah functional. We now present some definitions and results that are very useful in a discrete context. We stick to the notation that I introduced in [4], in order to mantain coherence between these two works.
Definition 1.4.1 (P Cn(0, 1)). We define P Cn(0, 1) as the set of all functions
u : [0, 1] → R such that u is constant in
i n, i + 1 n for every i = 0, . . . , n − 1 .
In particular P Cn(0, 1) ⊆ L2(0, 1). For simplicity, we also ask that u is continuous in 0. We define ui:= u i n for i = 1, . . . , n, hi := ui+1− ui for i = 1, . . . , n − 1.
In the sequel, we often call “heights” of the function u the values hi for i = 1, . . . , n − 1.
Definition 1.4.2 (Discrete derivative). For every function u : [0, 1] → R, let us define
˜ u : R → R as follows ˜ u (x) := u (x) if x ∈ [0, 1] , ˜ u (x) := u (0) if x < 0, ˜ u (x) := u (1) if x > 1.
The discrete derivative of step 1n of u is defined as
D±n1u (x) := ˜ ux ±1n− ˜u (x) ±1 n .
We point out that if u ∈ P Cn(0, 1), then D1nu|[0,1] is the function in P Cn(0, 1) such that Dn1u i= ui+1−ui 1 n = nhi if i = 1, . . . , n − 1, Dn1u n= 0.
In the sequel when we write D1nu we actually mean D
1
Theorem 1.4.3 (A discrete version of Arzel`a-Ascoli theorem). Let nk be a sequence of
natural numbers going to +∞ and unk be a sequence of functions such that (1) unk ∈ P Cnk(0, 1),
(2) unk(0) are uniformly bounded,
(3) D 1 nkun k L2(0,1)
are uniformly bounded.
Then, up to subsequences, there exists a function u∞ such that unk converges to u∞ uniformly on [0, 1].
1.5
General results
Theorem 1.5.1 (Arzel`a-Ascoli). Let (X, d) be a metric space and let un(t) be a sequence of functions in C[0, +∞) , X. Let us assume that the following hypotheses hold
(1) for every t ∈ [0, ∞) the sequenceun(t)
n∈N is relatively compact in X,
(2) for every A ∈ [0, +∞) there exists a constant CAsuch that for every 0 < t1< t2≤ A, for every n ∈ N it holds that
dun(t1) − un(t2)
≤ CA√t2− t1.
Then, up to subsequences, un(t) converges to some u∞(t) ∈ C
[0, +∞) , X uniformly on compact subsets of [0, +∞).
Now we state a slight modification of this theorem in the case in which the approxi-mating functions are not continuous, but still a property of “uniform discrete continuity” holds.
Theorem 1.5.2 (A compactness result for piecewise constant functions). Let (X, d) be
a metric space and let un(t) : [0, +∞) → X be a sequence of functions. Let us assume
that the following hypotheses hold
(1) for every t ∈ [0, +∞) the sequenceun(t)
n∈N is relatively compact in X,
(2) for every n ∈ N, for every k ∈ N the function un(t) is constant in h k n, k+1 n ,
(3) for every A ∈ [0, +∞) there exists a constant CA such that for every n ∈ N, for every integers k, h ≤ nA it holds that
d un k n − un h n ≤ CA s |k − h| n .
Then, up to subsequences, un(t) converges to some u∞(t) ∈ C
[0, +∞) , X uniformly on compact subsets of [0, +∞).
We point out that the two previous theorems do not require the completeness of the metric space (X, d).
Theorem 1.5.3 (Helly’s selection theorem). Let us consider a sequence of functions
ψn(x) : [0, +∞) → [a, b] such that every ψi is non increasing. Then there exists a
subsequence nk such that the sequence ψnk(x) admits a limit for every x ∈ [0, +∞). The limit function ψ∞(x) is obviously non increasing.
Now we present a specific result of differentiation under the integral sign that we need in the following chapters.
Theorem 1.5.4. Let us consider the two-dimensional domain [0, T ] × [0, 1]. The first
variable is denoted with t and the second one with x. Let us assume that
(1) h(t, x) ∈ L1t,x[0, T ] × [0, 1],
(2) there exists a negligible set E ⊆ [0, 1] such that for every x ∈ Ec the function
t → h(t, x) is in L1t(0, T ) (always true by Fubini’s theorem) and admits an L1t(0, T ) weak time derivative, i.e. ∂th(t, x) ∈ L1
t(0, T ).
Then the L1t(0, T ) function
H(t) :=
ˆ x=1 x=0
h(t, x) dx
admits a weak derivative that belongs to L1t(0, T ) and is given by
∂t ˆ x=1 x=0 h(t, x) dx ! = ˆ x=1 x=0 (∂th)(t, x) dx.
Proof. We want to prove that
∂tH(t) = L(t), where L(t) := ˆ x=1 x=0 (∂th)(t, x) dx.
Hence the thesis is that for every η ∈ Cc∞(0, T ) it holds that ˆ t=T t=0 η(t)L(t) dt = − ˆ t=T t=0 η0(t)H(t) dt, that is ˆ t=T t=0 η(t) "ˆ x=1 x=0 (∂th)(t, x) dx # dt = − ˆ t=T t=0 η0(t) "ˆ x=1 x=0 h(t, x) dx # dt.
But now − ˆ t=T t=0 η0(t) "ˆ x=1 x=0 h(t, x) dx # dt = − ˆ x=1 x=0 "ˆ t=T t=0 η0(t)h(t, x) dt # dx = − ˆ x=1 x=0 "ˆ t=T t=0 −η(t)∂th(t, x) dt # dx (1.5.1) = ˆ t=T t=0 η(t) "ˆ x=1 x=0 ∂th(t, x) dx # dt = ˆ t=T t=0 η(t)L(t) dt,
where in (1.5.1) we have used hypothesis (2) on the set of full measure Ec. Therefore the proof is complete.
2
General theory of Gradient Flows
in metric spaces
2.1
Definition of descending metric slope and main
prop-erties
First of all we need to recall the definition of descending metric slope and its main properties. We follow the same path of [4].
Definition 2.1.1 (Descending metric slope). Let (X, d) be a metric space and let F :
X → R be a function. Let us consider any x0 ∈ X such that F (x0) ∈ R. The metric
slope of F at x0 is defined as |∇F |(x0) := lim sup r→0+ F (x0) − inf {F (x) : d(x, x0) ≤ r} r . We also define |∇F |(x0) := +∞ if F (x0) = +∞.
Clearly it holds that |∇F |(x0) is always greater than or equal to 0 (including +∞), because x0 is a competitor for the infimum.
An equivalent definition, in the case in which x0 is not an isolated point of X, is the following |∇F |(x0) = lim sup x→x0 max {F (x0) − F (x), 0} d(x, x0) .
Proposition 2.1.2. If X = H is an Hilbert space and F : H → R is a C1 function, then the metric slope coincides with the norm of the gradient, i.e.
|∇F | (x) = k∇F (x)kH.
We omit the proof, that the reader can find in [4], Chapter 4, Lemma 4.1.2 in the particular case of H = Rn. It is however easy to generalize the proof to any Hilbert space.
Another question that I addressed in [4] concerns the connections between the Gamma-limit of the functions and the Gamma-Gamma-limit of the slopes.
The strongest statement that one might conjecture Fn Γ → F∞ ⇒? |∇Fn| Γ → |∇F∞|
has in general a negative answer, and the question is also ill-posed, since the Γ-limit of slopes might not exist.
The best result that one can obtain under sufficiently general assumption, as far as we know, is the following.
Theorem 2.1.3. Let (X, d) be a metric space and let us consider a sequence of functions
Fn: X → R. Let us assume that
• the sequence Fn Gamma-converges to F∞ (Gamma-convergence),
• for every n in N the function Fnis lower semicontinuous (Lower semicontinuity),
• for every r > 0, x0 ∈ X, M ∈ R there exists a compact set K ⊆ X such that
{x ∈ X : d(x, x0) ≤ r, Fn(x) ≤ M } ⊆ K for every n ∈ N
(Local equicoerciveness).
Then it holds that
Γ − lim sup
n→+∞
|∇Fn|(x) ≤ |∇F∞|(x).
The proof can be found in [4], in which it is also shown that none of the hypotheses can be removed without making the statement false.
Moreover, it is not possible to improve this theorem in order to deduce from the same hypotheses also the existence of recovery sequences for the Γ-limsup of slopes with bounded energy. A counterexample to this statement is exposed in [4], Chapter 4, Example 4.2.6.
2.2
Absolutely Continuous curves in metric spaces
Definition 2.2.1 (Absolutely Continuous curves in a metric space). Let (X, d) be a
metric space and let p ∈ (1, +∞) be a real number. The set of p-Absolutely Continuous curves in X is defined as ACp(X) :=nu : [0, +∞) → X s.t. ∃g ∈ Lploc(0, +∞) s.t. ∀ 0 ≤ τ0 < τ1 < +∞ du(τ1), u(τ0) ≤ ˆ τ1 τ0 g(s) ds ) .
Sometimes, when we want to stress the time domain, we also write ACp[0, +∞) , X or ACp[τ
0, τ1] , X
.
We point out that the previous definition implies that every Absolutely Continuous curve is continuous.
Theorem 2.2.2. If u belongs to ACp(X), then there exists a function gu ∈ Lploc(0, +∞) such that • ∀ 0 ≤ τ0< τ1 < +∞ it holds that du(τ1), u(τ0) ≤ ˆ τ1 τ0 gu(s) ds,
• for every g that satisfies the same property, it holds that
gu(s) ≤ g(s) for almost every s ∈ [0, +∞),
• for almost every s0∈ [0, +∞) it holds that
gu(s0) = lim
h→0
du(s0+ h), u(s0)
|h| .
For these reasons, the function gu(s) is called metric derivative of u and it is often denoted by | ˙u| (s).
We omit the proof, that the reader can find in [2].
Lemma 2.2.3. Let us consider a sequence of curves un∈ ACp
[τ0, τ1] , Xand a curve
u∞: [τ0, τ1] → X. Let us suppose that
• un(t) converges to u∞(t) for every t ∈ [τ0, τ1],
• | ˙un| (t) * g∞(t) with respect to the weak Lp topology in [τ0, τ1].
Then u∞(t) belongs to ACp
[τ0, τ1] , Xand | ˙u∞| (t) ≤ g∞(t).
The proof is an immediate consequence of the definition of Absolutely Continuous curves and the lower semicontinuity of the norm with respect to the weak convergence.
From now on we focus our attention on the special case in which (X, d) = Lp(0, 1) and we use the notation u(t, x) for u(t)(x).
Theorem 2.2.4. Let us consider
u ∈ ACp[τ0, τ1] , Lp(0, 1).
Let g(t) denote the metric derivative of u. Then
• the function u belongs to Lp(τ
0, τ1) × (0, 1)
,
• the function u admits a weak time derivative, i.e. there exists F ∈ Lp(τ
0, τ1) × (0, 1)
such that for every ψ ∈ Cc∞(τ0, τ1) × (0, 1)it holds that ¨ (τ0,τ1)×(0,1) u(t, x)∂ψ ∂t(t, x) dt dx = − ¨ (τ0,τ1)×(0,1) F (t, x)ψ(t, x) dt dx,
• k∂tuk Lp(τ
0,τ1)×(0,1)
≤ kgkLp(τ
0,τ1).
Proof. Step 0: The function u is measurable in the couple (t, x). We omit the proof of
this fact, since we think that this standard result of measure theory is not very interesting in our context.
Step 1: We show that u belongs to Lp(τ0, τ1) × (0, 1)
. ¨ [τ0,τ1]×[0,1] |u(t, x)|p dx dt = ˆ t=τ1 t=τ0 "ˆ x=1 x=0 |u(t, x)|p dx # dt = ˆ t=τ1 t=τ0 ku(t)kpLp(0,1) dt ≤ ˆ t=τ1 t=τ0 h
ku(t) − u(τ0)kLp(0,1)+ ku(τ0)kLp(0,1)
ip dt ≤ 2p−1 ˆ t=τ1 t=τ0 ku(t) − u(τ0)kpLp(0,1) dt + 2p−1(τ1− τ0) ku(τ0)kpLp(0,1). (2.2.1)
By hypothesis we know that
du(t), u(τ0)≤ ˆ t τ0 g(s) ds, du(t), u(τ0) p ≤ ˆ t τ0 g(s) ds p , ˆ x=1 x=0 |u(t, x) − u(τ0, x)|p ≤ ˆ t τ0 g(s) ds p , ˆ x=1 x=0 |u(t, x) − u(τ0, x)|p ≤ (t − τ0)(p−1)· ˆ t τ0 |g(s)|p ds. (2.2.2)
Combining (2.2.1) and (2.2.2) we deduce that ¨ [τ0,τ1]×[0,1] |u(t, x)|p dx dt ≤ ≤ 2p−1 ˆ t=τ1 t=τ0 " (t − τ0)p−1 ˆ t τ0 |g(s)|p ds # dt + (τ1− τ0) ku(τ0)kpLp(0,1) ! ≤ 2p−1 ˆ t=τ1 t=τ0 (t − τ0)p−1kgkpLp(τ 0,τ1) dt + (τ1− τ0) ku(τ0)k p Lp(0,1) ! .
Since τ0 < τ1 < +∞, the right-hand side of the last expression is finite, so we have
shown that u belongs to Lp(τ0, τ1) × (0, 1).
Step 2: We show that for every ε, for every 0 < |h| < ε it holds that
¨ (τ0+ε,τ1−ε)×(0,1) |u(t + h, x) − u(t, x)| h p dt dx ≤ kgkpLp(τ 0,τ1). (2.2.3)
In order to simplify the notation, we assume that h is positive (nothing changes for negative values of h). We know that ˆ t=τ1−ε t=τ0+ε 1 hp ˆ x=1 x=0 |u(t + h, x) − u(t, x)|p dx dt ≤ ≤ ˆ t=τ1−ε t=τ0+ε 1 hp " h(p−1)· ˆ t+h t |g(s)|p ds # dt (2.2.4) = 1 h ˆ s=τ1−ε s=τ0+ε "ˆ t=s t=s−h |g(s)|p dt # ds = 1 h ˆ s=τ1−ε s=τ0+ε h · |g(s)|p ds = ˆ s=τ1−ε s=τ0+ε |g(s)|p ds ≤ kgkpLp(τ 0,τ1). (2.2.5)
where (2.2.4) follows from (2.2.2). Since (2.2.5) is exactly (2.2.3), Step 2 is complete.
Step 3: We show that u admits a weak time derivative.
Let us fix a small parameter ε > 0 and let us consider any sequence hn → 0+ such
that hn≤ ε for every n. We consider the family of functions
zn(t, x) :=
u(t + hn, x) − u(t, x)
hn
,
defined for t ∈ (τ0+ ε, τ1− ε).
In Step 2 we have shown that the sequence zn is uniformly bounded with respect to the norm of Lp(τ0+ ε, τ1− ε) × (0, 1), hence, up to subsequences, there exists a function F ∈ Lp(τ0+ ε, τ1− ε) × (0, 1)
such that
zn* F.
This means in particular that for every function ψ ∈ Cc∞(τ0+ 2ε, τ1− 2ε) × (0, 1)
lim n→+∞ ˆ 1 0 ˆ τ1−ε τ0+ε zn(t, x)ψ(t, x) dt dx = ˆ 1 0 ˆ τ1−ε τ0+ε F (t, x)ψ(t, x) dt dx. (2.2.6)
Moreover, by the lower semicontinuity of the norm in Lp with respect to the weak convergence, we know that
kF k
Lp(τ
0+ε,τ1−ε)×(0,1)
≤ kgkLp(τ
We observe that ˆ 1 0 ˆ τ1−ε τ0+ε zn(t, x)ψ(t, x) dt dx = = 1 hn ˆ 1 0 ˆ τ1−ε τ0+ε h u(t + hn, x) − u(t, x) i ψ(t, x) dt dx = 1 hn ˆ 1 0 (ˆ τ 1−ε τ0+ε+hn u(t, x)hψ(t − hn, x) − ψ(t, x) i dt + rn(x) ) dx, where rn(x) = ˆ τ1−ε+hn τ1−ε u(t, x)ψ(t − hn, x) dt,
and this is identically equal to zero, since we are assuming that ψ vanishes for t ∈ [τ1− 2ε, τ1− ε]. So ˆ 1 0 ˆ τ1−ε τ0+ε zn(t, x)ψ(t, x) dt dx = 1 hn ˆ 1 0 ˆ τ1−ε τ0+ε+hn u(t, x)hψ(t − hn, x) − ψ(t, x) i dt dx. (2.2.8) We take the limit as n approaches infinity and we deduce that
lim n→+∞ ˆ 1 0 ˆ τ1−ε τ0+ε zn(t, x)ψ(t, x) dt dx = − ˆ 1 0 ˆ τ1−ε τ0+ε+hn u(t, x)∂tψ(t, x) dt dx (2.2.9) = − ˆ 1 0 ˆ τ1−ε τ0+ε u(t, x)∂tψ(t, x) dt dx, (2.2.10)
where (2.2.9) follows by the dominated convergence theorem on the right-hand side of (2.2.8) and (2.2.10) follows from the assumption that ψ vanishes for t ∈ [τ0+ ε, τ0+ 2ε].
Recalling (2.2.6), we have proved that ˆ 1 0 ˆ τ1−ε τ0+ε F (t, x)ψ(t, x) dt dx = − ˆ 1 0 ˆ τ1−ε τ0+ε u(t, x)∂tψ(t, x) dt dx
for every function ψ ∈ Cc∞(τ0+ 2ε, τ1− 2ε) × (0, 1)
.
Since ε is arbitrary and we have the uniform estimate (2.2.7), this implies indeed that the function u admits a weak time derivative for t ∈ (τ0, τ1).
Corollary 2.2.5. If u(t, x) ∈ ACp[τ0, τ1] , Lp(0, 1)
, then
(1) The family (indexed by h) of Lp[0, 1] × [τ0, τ1)
functions
u(t + h, x) − u(t, x) h
converges strongly to ∂tu(t, x) as h → 0 in the Lp[0, 1] × [τ0, τ1)
topology.
In particular, translating this result on the one-dimensional sections, we get the following results.
(2) There exists an appropriate subsequence hn → 0 such that for almost every x ∈
[0, 1] the sequence of Lp(τ0, τ1) functions
u(t + hn, x) − u(t, x)
hn
converges strongly to ∂tu(t, x) as n → +∞ in the Lp(τ0, τ1) topology.
Therefore for almost every x ∈ [0, 1] the function t → u(t, x) belongs to W1,p(τ0, τ1)
(in particular it is continuous).
(3) There exists an appropriate subsequence hn → 0 such that for almost every τ ∈
[τ0, τ1] the sequence of Lp(0, 1) functions
u(τ + hn, x) − u(τ, x)
hn
converges strongly to ∂tu(τ, x) as n → +∞ in the Lp(0, 1) topology.
Proof. The proof of the convergence in the couple (t, x) follows from the proof of Theorem
2.2.4 and from the general fact that the difference quotients converge weakly if and only if they converge strongly (see for example [9], Proposition 9.3), so (1) is proved.
Now we want to deduce theses (2) and (3) from (1). We notice that, in general, if we know that
Gn(t, x) → 0 in the Lp [0, 1] × [τ0, τ1] topology, i.e. ˆ x=1 x=0 ˆ t=τ1 t=τ0 |Gn(t, x)|p dt dx → 0,
then by Fubini’s theorem we know that ˆ t=τ1
t=τ0
|Gn(t, x)|p dt → 0 in the L1(0, 1) topology.
We recall that the L1 convergence implies convergence almost everywhere (on a subse-quence). Therefore, up to subsequences, we deduce that for almost every x ∈ [0, 1]
ˆ t=τ1
t=τ0
|Gn(t, x)|p dt → 0,
so for almost every x ∈ [0, 1] the functions Gn(t, x) converge to zero in the Lp(τ0, τ1)
topology.
Using this fact with
Gn(t, x) :=
u(t + hn, x) − u(t, x)
hn
we deduce thesis (2).
Corollary 2.2.6. If u(t, x) ∈ ACp[τ0, τ1] , Lp(0, 1), then for almost every s0 ∈ [τ0, τ1] | ˙u| (s0) = ˆ x=1 x=0 |∂tu(s0, x)|p dx !1p . (2.2.11)
Proof. First of all, we compute the following estimate for h positive.
d(u(s 0+ h), u(s0)) |h| p = 1 hp ˆ x=1 x=0 |u(s0+ h, x) − u(s0, x)|p dx = 1 hp ˆ x=1 x=0 ˆ s0+h s0 ∂tu(s, x) ds p dx (2.2.12) ≤ 1 h ˆ x=1 x=0 ˆ s0+h s0 |∂tu(s, x)|p ds dx (2.2.13) = 1 h ˆ s0+h s0 "ˆ x=1 x=0 |∂tu(s, x)|p dx # ds, (2.2.14)
where in (2.2.13) we have used H¨older’s inequality. Passage (2.2.12) is well posed thanks to Theorem 2.2.4.
Let us assume that
s0 is a Lebesgue point for the function s → ˆ x=1
x=0
|∂tu(s, x)|p dx (2.2.15)
(if we apply Fubini’s theorem to the thesis of Theorem 2.2.4, we deduce that this function belongs to L1(τ0, τ1), therefore almost every point is a Lebesgue point).
In this case, we compute the limsup in (2.2.14) and we get that
lim sup h→0+ d(u(s 0+ h), u(s0)) |h| p ≤ lim sup h→0+ 1 h ˆ s0+h s0 "ˆ x=1 x=0 |∂tu(s, x)|p dx # ds = lim h→0+ 1 h ˆ s0+h s0 "ˆ x=1 x=0 |∂tu(s, x)|p dx # ds = "ˆ x=1 x=0 |∂tu(s0, x)|p dx # . (2.2.16)
Moreover, for every s0 it holds that
lim inf h→0+ d(u(s0+ h), u(s 0)) |h| p = lim inf h→0 ´x=1 x=0 |u(s0+ h, x) − u(s0, x)| p dx hp = lim inf h→0+ ˆ x=1 x=0 |u(s 0+ h, x) − u(s0, x)| h p dx ≥ ˆ x=1 x=0 lim inf h→0+ |u(s 0+ h, x) − u(s0, x)| h p dx (2.2.17) = ˆ x=1 x=0 |∂tu(s0, x)|p dx, (2.2.18)
where in (2.2.17) we have used Fatou’s Lemma.
Combining (2.2.16) and (2.2.18) we deduce that for almost every s0
lim h→0+ d(u(s0+ h), u(s0)) |h| = ˆ x=1 x=0 |∂tu(s0, x)|p dx !1 p .
Now we can repeat all the previous passages for h negative. Recalling that for almost every s0∈ [τ0, τ1] it holds that
| ˙u| (s0) = lim
h→0
d(u(s0+ h), u(s0))
|h|
by Theorem 2.2.2, the proof is complete.
Proposition 2.2.7. Let w(t, x) ∈ AC2[τ0, τ1] , L2(0, 1)
. Then the function h(t, x) :=
w(t, x)2 satisfies all the assumptions of Theorem 1.5.4.
Proof. We have to check that
(1) w(t, x)2 ∈ L1 t,x [τ0, τ1] × [0, 1] ,
(2) there exists a negligible set E ⊆ [0, 1] such that for every x ∈ Ec the function
t → w(t, x)2 is in L1t(τ0, τ1) (always true by Fubini’s theorem) and admits an
L1t(τ0, τ1) weak time derivative, i.e. ∂t w(t, x)2∈ L1t(τ0, τ1).
Now, by Theorem 2.2.4 we know that w(t, x) belongs to L2t,x[τ0, τ1] × [0, 1]
, hence hypothesis (1) is satisfied. In particular (by Fubini’s Theorem) there exists a negligible set E1⊆ [0, 1] such that for every x ∈ E1cthe function w(t, x) belongs to L2t(τ0, τ1).
By Theorem 2.2.4 we also know that there exists a negligible set E2 ⊆ [0, 1] such that for every x ∈ E2c the function t → w(t, x) belongs to W1,2(τ0, τ1). Let us call ∂tw(t, x)
its derivative, which we know that is an L2t(τ0, τ1) function.
It is now clear that, for the negligible set E := E1∪ E2, the function t → w(t,x)2 be-longs to W1,1(τ0, τ1), in fact it is in L1t(τ0, τ1) and its partial derivative is 2w(t,x)∂tw(t, x),
which belongs to L1t(τ0, τ1) since it is a product of two L2t(τ0, τ1) functions.
Corollary 2.2.8. If w(t, x) ∈ AC2[τ0, τ1] , L2(0, 1)
, then the L1t(τ0, τ1) function
H(t) :=
ˆ x=1 x=0
w(t, x)2dx
admits a weak derivative that belongs to L1t(τ0, τ1) and is given by
∂t ˆ x=1 x=0 w(t, x)2dx ! = ˆ x=1 x=0 2w(t, x)(∂tw(t, x)) dx.
However, one should not expect too much from Absolutely Continuous curves, as the following example shows.
Example 2.2.9. Let us define the function z as follows
z(x) :=X
i∈N
ci1Ii(x),
where ci are real numbers and Iiare pairwise disjoint measurable subsets of [0, 1]. If the
following condition holds
X i∈N
L (Ii)c2i < +∞,
then z ∈ L2(0, 1).
Let us now define u(t, x) := tz(x). It is clear that u(t) ∈ AC2[0, +∞) , L2(0, 1). In fact ku(τ1), u(τ0)kL2(0,1) = |τ1− τ0| kz(x)kL2(0,1) ≤ ˆ τ1 τ0 g(s) ds,
where g(s) ≡ kz(x)kL2(0,1) for every s, so g clearly belongs to L2loc(0, +∞).
For t = 0, u(0, x) is the null function, that satisfies every possible request of regularity. However, for every ε > 0, u(ε, x) has at least a countable number of jumps, or even more, depending on the sets Ii.
2.3
Curves of Maximal Slope in metric spaces
Let us consider a Hilbert space H and a C1 function F : H → R. Let us assume that
v(t) : [0, +∞) → H is a C1 curve. We estimate − d dtF (v(t)) = − h∇F (v(t)), ˙v(t)iH (2.3.1) ≤ k∇F (v(t))kH · k ˙v(t)kH (2.3.2) ≤ 1 2k∇F (v(t))k 2 H + 1 2k ˙v(t)k 2 H, (2.3.3)
where in (2.3.1) we have just used the chain rule (we recall that F is C1). Therefore − d dtF (v(t)) ≤ 1 2k∇F (v(t))k 2 H+ 1 2k ˙v(t)k 2
H for every v(t) regular. (2.3.4)
Now we want to understand under which assumptions the previous chain of inequal-ities is in fact a chain of equalinequal-ities. Since (2.3.2) is an equality if and only if
∇F (v(t)) = λt˙v(t) for some λtnon positive
and (2.3.3) is an equality if and only if
then − d dtF (v(t)) = 1 2k∇F (v(t))k 2 H + 1 2k ˙v(t)k 2 H if and only if ∇F (v(t)) = − ˙v(t). (2.3.5)
Combining (2.3.4) and (2.3.5) we deduce that
(
F and v(t) are of class C1,
−dtdF (v(t)) ≥ 12k∇F (v(t))k2H +12k ˙v(t)k2H, =⇒ ( − d dtF (v(t)) = 1 2k∇F (v(t))k 2 H +12k ˙v(t)k 2 H, ∇F (v(t)) = − ˙v(t). (2.3.6)
These considerations, that close the matter in such regular cases, motivate the defi-nitions that follow, which extend the concept of Gradient Flow in a weaker context.
The Hilbert space is replaced by a metric space and we have weaker assumptions on the function F .
We give three definitions of Maximal Slope Curve in the spirit of (2.3.6), but using an integral formulation. As it is well known, integral formulations make sense in more general contexts than differential formulations, and inequalities are more robust than equalities.
Definition 2.3.1 (Pure Maximal Slope Curve). Let (X, d) be a metric space and F :
X → R be a function. A Pure Maximal Slope Curve (PMSC) for the function F in the metric space X is a curve u(t) : [0, +∞) → X such that
• u(t) ∈ AC2[0, +∞) , X,
• for every 0 ≤ t1 ≤ t2 it holds that
F (u(t1)) − F (u(t2)) ≥ 1 2 ˆ t2 t1 |∇F |2(u(t)) dt + 1 2 ˆ t2 t1 | ˙u|2(t) dt.
Definition 2.3.2 (Semi Pure Maximal Slope Curve). Let (X, d) be a metric space and
F : X → R be a function. A Semi Pure Maximal Slope Curve (SPMSC) for the function F in the metric space X is a triple (u(t), E, ψ), where
• u(t) ∈ AC2[0, +∞) , X,
• E ⊆ (0, +∞) is negligible,
• ψ : [0, +∞) → R is non increasing,
• for every t ∈ Ec it holds that ψ(t) = F (u(t)),
• for every 0 ≤ t1 ≤ t2 it holds that
ψ(t1) − ψ(t2) ≥ 1 2 ˆ t2 t1 |∇F |2(u(t)) dt +1 2 ˆ t2 t1 | ˙u|2(t) dt.
The reason why we ask that 0 does not belong to E is to make sense properly of the initial datum (otherwise some pathologies might arise, as we show in Chapter 5).
However, this well posedness of the initial condition does not pass to the limit, as we show in Theorem 2.3.12.
It is therefore useful to give a third definition, in which it is not requested that
ψ(0) = F (u(0)). We will see that this definition is the most appropriate when passing
to the limit.
Definition 2.3.3 (Weak Maximal Slope Curve). Let (X, d) be a metric space and F :
X → R be a function. A Weak Maximal Slope Curve (WMSC) for the function F in the metric space X is a triple (u(t), E, ψ), where
• u(t) ∈ AC2[0, +∞) , X,
• E ⊆ [0, +∞) is negligible,
• ψ : [0, +∞) → R is non increasing,
• for every t ∈ Ec it holds that ψ(t) = F (u(t)),
• for every 0 ≤ t1 ≤ t2 it holds that
ψ(t1) − ψ(t2) ≥ 1 2 ˆ t2 t1 |∇F |2(u(t)) dt +1 2 ˆ t2 t1 | ˙u|2(t) dt.
It is clear that every Pure Maximal Slope Curve is a Semi Pure Maximal Slope Curve (it is sufficient to consider E = ∅) and that every Semi Pure Maximal Slope Curve is a Weak Maximal Slope Curve.
In the sequel, we often say that u(t) is a Weak/Semi Pure Maximal Slope Curve meaning really that there exists a negligible set E ⊆ [0, +∞) and a non increasing function ψ : [0, +∞) → R such that the triple (u(t), E, ψ) is a Weak/Semi Pure Maximal Slope Curve.
Now we state the first result that we can expect having (2.3.6) in mind.
Proposition 2.3.4. If the metric space (X, d) is a Hilbert space and the function F
is Cloc1,1, then for every initial condition x0 there exists a unique Weak Maximal Slope
Curve with initial datum x0, and this curve is the unique solution to the Gradient Flow
of F
(
˙
u(t) = −∇F (u(t)), u(0) = x0,
where obviously ∇F denotes the gradient of F as a vector in H. Moreover, every Weak Maximal Slope Curve is a Pure Maximal Slope Curve.
Remark 2.3.5. If F is a lower semicontinuous function and (u(t), E, ψ) is a Semi Pure
Maximal Slope Curve for F , then F (u(0)) ≥ F (u(t)) for every t.
Proof. Since E is negligible, it is possible to find a sequence tnsuch that
• limn→+∞tn= t.
By definition, we know that
F (u(0)) − F (u(tn)) = ψ(0) − ψ(tn)
≥ 0. (2.3.7)
We also know that
lim inf n→+∞F (u(tn)) ≥ F lim n→+∞u(tn) (2.3.8) = F (u(t)), (2.3.9)
where in (2.3.8) we have used the lower semicontinuity of F and in (2.3.9) we have used the continuity of u(t), that belongs to AC2(X).
Now we take the limsup in the left-hand side of (2.3.7) and we use (2.3.9) to gain that
F (u(0)) − F (u(t)) ≥ 0,
as desired.
We point out that the previous result fails in the case of Weak Maximal Slope Curves, in which it is possible that F (u(0)) < F (u(t)), at least for small times t. For a concrete example, we refer to Chapter 5, Section 5.2.3.
Remark 2.3.6. If F is a continuous function and (u(t), E, ψ) is a Weak Maximal Slope
Curve for F , then the function t → F (u(t)) is non increasing.
Proof. Let us consider any 0 ≤ s < t. Since E is negligible, it is possible to find two
sequences sn, tn such that
• sm< tk for every m ∈ N, for every k ∈ N,
• sn∈ Ec for every n ∈ N,
• tn∈ Ec for every n ∈ N,
• limn→+∞sn= s,
• limn→+∞tn= t.
By definition, we know that
F (u(sn)) − F (u(tn)) = ψ(sn) − ψ(tn)
≥ 0. (2.3.10)
We also know that
lim n→+∞F (u(sn)) = F lim n→+∞u(sn) (2.3.11) = F (u(s)), (2.3.12)
where in (2.3.11) we have used the continuity of F and in (2.3.12) we have used the continuity of u(t), which we know that belongs to AC2(X).
Now we take the limit in the left-hand side of (2.3.10) and we use (2.3.12) to gain that
F (u(s)) − F (u(t)) ≥ 0,
as desired.
The previous remark can easily be generalized to the context in which F|Z is contin-uous (with respect to the distance d) and (u(t), E, ψ) is a Weak Maximal Slope Curve for F , that for some reason (for example by Remark 2.3.5) we know that lies in Z for every t ≥ 0.
We point out, however, that in the case (that we consider later on in this thesis) of X := L2(0, 1), Z := H1(0, 1), asking that F|Z is continuous means that
for every un⊆ H1(0, 1) s.t. un→ u, then F (uL2 n) → F (u)
and not that
for every un⊆ H1(0, 1) s.t. unH
1
→ u, then F (un) → F (u).
The problem is that we cannot change the distance d with a stronger one in the previous remark, because a priori we only know that t → u(t) is Absolutely Continuous with respect to the initial distance d.
Remark 2.3.7. If F is a continuous function and (u(t), E, ψ) is a Semi Pure Maximal
Slope Curve for F , then u(t) is a Pure Maximal Slope Curve for F .
Proof. We know that the function
f (t) := F (u(t))
defined for t ∈ [0, +∞) is continuous, because it is composition of two continuous maps. We know that ψ(t) is non increasing and coincides in 0 and almost everywhere with
f , that is continuous. This implies that ψ cannot have discontinuities and therefore ψ(t)
and F (u(t)) must coincide everywhere, so the proof is complete.
We point out that the requests in the previous remark cannot be weakened asking that F is only lower semicontinuous. A counterexample to this fact is shown in Chapter 5.
Proposition 2.3.8 (Invariance under isometries). Let us assume that(u(t), E, ψ) is a
Pure/Semi Pure/Weak Maximal Slope Curve for the function F in the metric space (X, d). Let us assume that u(t) ∈ Z for every t ≥ 0 for some subspace Z ⊆ X and that
f : (Z, d) → (Y, dY) is an isometry with inverse h. Let us define
e
F (u) := F (h(u)).
Then (f (u(t)), E, ψ) is a Pure/Semi Pure/Weak Maximal Slope Curve for the function
e
Proof. We point out that f (u(t)) is well defined, since we know that u(t) belongs to Z
for every t ≥ 0.
For simplicity, we only deal with Weak Maximal Slope Curves, the proof in the other cases is analogous. We know by hypothesis that
(H1) u(t) is in AC2[0, +∞), (X, d), (H2) E ⊆ [0, +∞) is negligible,
(H3) ψ : [0, +∞) → R is non increasing,
(H4) for every t ∈ Ec it holds that ψ(t) = F (u(t)),
(H5) for every 0 ≤ t1 ≤ t2 it holds that
ψ(t1) − ψ(t2) ≥ 1 2 ˆ t2 t1 |∇F |2(u(t)) dt +1 2 ˆ t2 t1 | ˙u|2(t) dt.
The thesis is that
(T1) f (u(t)) is in AC2[0, +∞) , (Y, dy)
,
(T2) E ⊆ [0, +∞) is negligible,
(T3) ψ : [0, +∞) → R is non increasing,
(T4) for every t ∈ Ec it holds that ψ(t) =F (f (u(t))),e
(T5) for every 0 ≤ t1 ≤ t2 it holds that
ψ(t1) − ψ(t2) ≥ 1 2 ˆ t2 t1 ∇Fe 2 (f (u(t))) dt + 1 2 ˆ t2 t1 ˙ f (u) 2 (t) dt.
(T2), (T3) and (T4) follow immediately from (H2), (H3) and (H4). Also (T1) and (T5) follow from (H1) and (H5) thanks to the fact that
dX(u, v) = dZ(u, v) = dY (f (u), f (v)) .
Now we want to state and prove a theorem concerning the existence of Semi Pure Maximal Slope Curves. The proof follows the idea of minimizing movements. Roughly speaking, the idea is to consider a discretization in time of step n1 and to ask inductively that uk+1n = x, where x is one of the minimizers of the function
F (x) + n 2d x, u k n 2 . (2.3.13)
As n approaches infinity, we show that this curve converges to a a Semi Pure Maximal Slope Curve, as desired.
The basic idea for which the previous formula is in fact an approximation of the Gradient Flow is clear if we assume to be in a Hilbert space and to have some hypotheses of regularity. Indeed, if we compute the derivative of (2.3.13) at point x, thanks to the request of minimality, we get that
0 = ∇ " F (x) + n 2d x, u k n 2# |x=x = ∇F (x) + n x − u k n , that is to say uk+1n − uk n 1 n = −∇F u k + 1 n .
The previous equation is the Euler backward approximation of step n1 of the Gradient Flow equation
d
dt(u(t)) = −∇F (u(t)).
Theorem 2.3.9 (Existence of Semi Pure Maximal Slope Curves). Let (X, d) be a metric
space, let F : X → R ∪ {+∞} be a function and let x0 be an element of X. Let us assume
that
(1) F (x0) ∈ R, (2) inf F > −∞,
(3) F is lower semicontinuous and locally coercive, i.e. for every x0 ∈ X, for every
r > 0, for every M in R the set {x s.t. F (x) ≤ M } ∩ B(x0, r) is compact,
(4) for every sequence xn⊆ X such that xn→ x∞, |F (xn)| ≤ M, |∇F | (xn) ≤ M it holds that ( limn→+∞F (xn) = F (x∞), lim infn→+∞|∇F | (xn) ≥ |∇F | (x∞).
Then there exists a Semi Pure Maximal Slope Curve (u(t), E, ψ) such that u(0) = x0.
Proof. Step 1: We define the curve u(t).
We consider any sequence of curves un: [0, +∞) → X that satisfy the following inductive
requests
• un|[k n,
k+1
n ) is constant for every k ∈ N, • unk+1
n
= x, where x is one of the minimizers of the function
x → F (x) + n 2d x, un k n 2 .
We observe that the point x is well defined thanks to hypothesis (3).
Of course there might be more than one minimizer for the function
x → F (x) +n 2d x, un k n 2 ,
therefore the sequence un is not uniquely identified. In the sequel we consider any sequence un with the above properties.
We notice that the function F is non increasing along un, i.e.
F un k n ≥ F un k + 1 n for every n, k ∈ N. In particular it holds that
M := F (x0) ≥ F (un(t)) for every n ∈ N, for every t ∈ [0, +∞). (2.3.14)
By definition of un k+1 n it holds that F un k + 1 n +n 2d un k + 1 n , un k n 2 ≤ F un k n + 0, d un k + 1 n , un k n ≤ s F un k n − F un k + 1 n 2 n,
therefore by triangular inequality
d un i n , un h n ≤ r 2 n i−1 X k=h s F un k n − F un k + 1 n ≤ r 2 n √ i − h v u u t i−1 X k=h F un k n − F un k + 1 n ≤ r 2 n √ i − h s F un h n − F un i n ≤ r 2 n √ i − h√M − m, (2.3.15)
where m denotes the infimum of F , that we know to be finite thanks to hypothesis (2). Let us fix A ∈ [0, +∞) and let us consider I := [0, A]. From the previous estimate we deduce that for every n ∈ N, for every t ∈ I it holds that
dun(t), x0 ≤ r 2 n √ nA√M − m = q 2A(M − m).
