sign in sign up
<<
Home , Total variation, Willmore energy

Calculus of Variations we consider a Z I. Fonseca and G. Leoni u ∈ X 7→ F (u) := f(x, u(x), ∇u(x)) dx, (1) Carnegie Mellon University, USA Ω where X is a space (usually a Lp space or a Sobolev-type space), u :Ω → Rd, with Ω ⊂ RN 1 History an open , N and d are positive integers, and the density is a function f(x, u, ξ), with (x, u, ξ) ∈ The is a branch of mathe- Ω × Rd × Rd×N . Here, and in what follows, ∇u matical analysis that studies extrema and critical stands for the d × N matrix-valued distributional points of functionals (or energies). Here, by func- derivative of u. tional we mean a mapping from a function space The calculus of variations is a vast theory and to the real numbers. here we chose to highlight some contemporary as- One of the first questions that may be framed pects of the field, and we conclude this article by within this theory is Dido’s isoperimetric prob- mentioning a few forefront areas of application lem (see Subsection 2.3): to find the shape of a that are driving current research. curve of prescribed perimeter that maximizes the area enclosed. Dido was a Phoenician princess who emigrated to North Africa and upon arrival 2 Extrema obtained from the native chief as much territory In this section we address fundamental minimiza- as she could enclose with an ox hide. She cut the tion problems and relevant techniques in the cal- hide into a long strip, and used it to delineate the culus of variations. In geometry, the simplest ex- territory later known as Carthage, bounded by a ample is the problem of finding the curve of short- straight coastal line and a semi-circle. est length connecting two points, a geodesic.A It is commonly accepted that the systematic (continuous) curve joining two points A, B ∈ Rd development of the theory of the calculus of vari- is represented by a (continuous) function γ : ations began with the brachistochrone curve prob- [0, 1] → Rd such that γ(0) = A, γ(1) = B, and lem proposed by Johann Bernoulli in 1696: con- its length is given by sider two points A and B on the same vertical n plane but on different vertical lines. Assume that n X o A is higher than B, and that a particle M is mov- L(γ) := sup |γ (ti) − γ (ti−1)| , ing from A to B along a curve and under the ac- i=1 tion of gravity. The curve that minimizes the time where the supremum is taken over all partitions travelled by M is called the brachistochrone. The 0 = t0 < t1 < ··· < tn = 1, n ∈ N, of the interval solution to this problem required the use of in- R 1 0 [0, 1]. If γ is smooth, then L(γ) = 0 |γ (t)| dt. finitesimal calculus and was later found by Jacob In the absence of constraints, the geodesic is the Bernoulli, Newton, Leibniz and de l’Hˆopital.The straight segment with endpoints A and B, and so arguments thus developed led to the development L(γ) = |A − B|. Often in applications the curves of the foundations of the calculus of variations by are restricted to lie on a given , e.g., a Euler. Important contributions to the subject are (in this case, the geodesic is the shortest attributed to Dirichlet, Hilbert, Lebesgue, Rie- great circle joining A and B). mann, Tonelli, Weierstrass, among many others. The common feature underlying Dido’s and the 2.1 Minimal Surfaces brachistochrone problems is that one seeks to maximize or minimize a functional over a class A minimal is a surface of least area among of competitors satisfying given constraints. In all those bounded by a given closed curve. The both cases the functional is given by an problem of finding minimal surfaces, called the of a density depending on an underlying field and Plateau problem, was first solved in three dimen- some of its derivatives, and this will be the pro- sions in the 1930’s by Douglas and by Rado, and totype we will adopt in what follows. Precisely, in the 1960’s several authors, including Almgren,

1 2

De Giorgi, Fleming and Federer, addressed it us- In the 1920’s it was shown by Blaschke and by ing geometric theoretical tools. This ap- Thomsen that the Willmore energy is invariant proach gives existence of solutions in a “weak under conformal transformations of R3. Also, the sense”, and their regularity is significantly more Willmore energy is minimized by , with involved. De Giorgi proved that minimal surfaces resulting energy value 4π. Therefore, W(S) − 4π are analytic except on a singular set of dimen- describes how much S differs from a sphere in sion at most N − 1. Later, Federer, based on terms of its bending. The problem of minimiz- earlier results by Almgren and Simons, improved ing the Willmore energy among the class of em- the dimension of the singular set to N − 8. The bedded tori T was proposed by Willmore, who sharpness of this estimate was confirmed with an conjectured in 1965 that W(T ) ≥ 2π2. This con- example by Bombieri, De Giorgi and Giusti. jecture has been proved by Marques and Neves in Important minimal surfaces are the so-called 2012. non-parametric minimal surfaces, which are given as graphs of real-valued functions. Precisely, 2.3 Isoperimetric Problems; the N given an Ω ⊂ R and a smooth func- Wulff set tion u :Ω → R, then the area of the graph of u, {(x, u(x)) : x ∈ Ω}, is given by The understanding of the surface structure of crystals plays a central role in many fields of Z physics, chemistry and materials science. If the F (u) := p1 + |∇u|2 dx. (2) Ω dimension of the crystals is sufficiently small, then the leading morphological mechanism is It can be shown that u minimizes the area of its driven by the minimization of surface energy. graph subject to prescribed values on the bound- Since the work of Herring in the 1950’s, a classi- ary of Ω if cal question in this field is to determine the crys- ! talline shape that has smallest surface energy for ∇u a given volume. Precisely, we seek to minimize div = 0 in Ω. p1 + |∇u|2 the surface integral Z ψ(ν(x)) dσ (3) 2.2 Willmore Functional ∂E

Recently many smooth surfaces, including tori, over all smooth sets E ⊂ RN with prescribed vol- have been obtained as minima or critical points ume, and where ν(x) is the outward unit nor- of certain geometrical functionals in the calculus mal to ∂E at x. The right variational framework of variations. An important example is the Will- for this problem is within the class of sets of fi- more (or bending) energy of a compact surface nite perimeter. The solution, which exists and S embedded in R3, namely the surface integral is unique up to translations, is called the Wulff R 2 k1+k2 W(S) := S H dσ, where H := 2 and k1 shape. A key ingredient in the proof is the Brunn- and k2 are the principal curvatures of S. This Minkowski inequality energy has a wide scope of applications, ranging N 1/N N 1/N N 1/N from materials science (e.g., elastic shells, bend- (L (A)) + (L (B)) ≤ (L (A + B)) ing energy), to mathematical biology (e.g., cell (4) membranes) to image segmentation in computer which holds for all Lebesgue measurable sets N vision (e.g., staircasing). A, B ⊂ R such that A+B is also Lebesgue mea- N Critical points of W are called Willmore sur- surable. Here L stands for the N-dimensional faces, and satisfy the Euler-Lagrange equation Lebesgue measure.

2 ∆SH + 2H(H − K) = 0, 3 The Euler Lagrange Equation where K := k1k2 is the and Consider the functional (1), in the scalar case 1 ∆S is the Laplace-Beltrami operator. d = 1, and where f of class C and X is the 3

Sobolev space X = W 1,p(Ω), 1 ≤ p ≤ +∞, of all Ω, then the variation u+tϕ is admissible if ϕ ≥ 0 functions u ∈ Lp(Ω) whose distributional gradi- and t ≥ 0. Therefore, the function g satisfies ent ∇u belongs to Lp(Ω; RN ). Let u ∈ X be a g0(0) ≥ 0, and the Euler-Lagrange equation (5) local minimizer of the functional F , that is, becomes the variational inequality

Z Z N Z  X ∂f ∂ϕ f(x, u(x), ∇u(x)) dx ≤ f(x, v(x), ∇v(x)) dx (x, u, ∇u) U U ∂ξ ∂x Ω i=1 i i for every open subset Ucompactly contained in ∂f  1,p + (x, u, ∇u)ϕ dx ≥ 0 Ω, and all v such that u − v ∈ W0 (U), where ∂u 1,p 1,p W0 (U) is the space of all functions in W (U) 1 “vanishing” on the boundary of ∂U. Note that for all nonnegative ϕ ∈ Cc (Ω). This is called the v will then coincide with u outside the set U. If obstacle problem, and the coincidence set {u = φ} 1 is not known a priori and is called the free bound- ϕ ∈ Cc (Ω) then u + tϕ, t ∈ R, are admissible, and thus ary. This is an example of a broad class of vari- ational inequalities and free boundary problems t ∈ R 7→ g(t) := F (u + tϕ) that have applications in a variety of contexts, including the modeling of the melting of ice (the has a minimum at t = 0. Therefore, under ap- Stefan problem), lubrication, and the filtration of propriate growth conditions on f, we have that a liquid through a porous medium. 0 g (0) = 0, i.e., A related class of minimization problems in

N which the unknowns are both an underlying field Z  X ∂f ∂ϕ (x, u, ∇u) u and a subset E of Ω, is the class of free dis- ∂ξ ∂x Ω i=1 i i continuity problems that are characterized by the ∂f  competition between a volume energy of the type + (x, u, ∇u)ϕ dx = 0. (5) (1) and a surface energy, e.g., as in (3). Impor- ∂u tant examples are in the study of liquid crystals, A function u ∈ X satisfying (5) is said to be a optimal design of composite materials in contin- weak solution of the Euler Lagrange equation as- uum mechanics (see Subsection 13.3), and image sociated to (1). segmentation in computer vision (see Subsection Under suitable regularity conditions on f and 13.4). u, (5) can be written in the strong form 5 Lagrange Multipliers ∂f div(∇ξf(x, u, ∇u)) = (x, u, ∇u), (6) ∂u The method of Lagrange multipliers in Banach spaces is used to find extrema of functionals G : where ∇ξf(x, u, ξ) is the gradient of the function f(x, u, ·). X → R subject to a constraint In the vectorial case d > 1 the same argument {x ∈ X : Ψ(x) = 0}, (7) leads to a system of partial differential equations (PDEs) in place of (5). where Ψ : X → Y is another functional and X and Y are Banach spaces. It can be shown that 1 4 Variational Inequalities, Free if G and Ψ are of class C and u ∈ X is an ex- Boundary and Free Discontinuity tremum of G subject to (7), and if the derivative DΨ(u): X → Y is surjective, then there exists Problems a continuous, linear functional λ : Y → R such We now add a constraint to the minimization that problem considered in the previous section. Pre- DG(u) + λ ◦ DΨ(u) = 0, (8) cisely, let d = 1 and let φ be a function in Ω. If where ◦ stands for the composition operator be- u is a local minimizer of (1) among all functions tween functions. The functional λ is called a La- v ∈ W 1,p(Ω) subject to the constraint v ≥ φ in grange multiplier. 4

In the special case in which Y = R, λ may be 7 Lower Semicontinuity identified with a scalar, still denoted by λ, and (8) takes the familiar form 7.1 The Direct Method DG(u) + λDΨ(u) = 0. The direct method in the calculus of variations provides conditions on the function space X and Therefore, candidates for extrema may be found on a functional G, as introduced in Section 5, that among all critical points of the family of function- guarantee the existence of minimizers of G. The als G + λΨ, λ ∈ R. method consists of the following steps: If G has the form (1) and X = W 1,p(Ω; d), R Step 1. Consider a minimizing sequence {un} ⊂ 1 ≤ p ≤ +∞, then typical examples of Ψ are X, i.e., limn→∞ G(un) = infu∈X G(u). Z Z Step 2. Prove that {un} admits a subsequence Ψ(u) := |u|s dx−c or Ψ(u) := u dx−c 1 2 {un } converging to some u0 ∈ X with respect Ω Ω k to some (weak) topology τ in X. When G has d for some constants c1 ∈ R, c2 ∈ R , and 1 ≤ s < an integral representation of the form (1), this is +∞. usually a consequence of a priori coercivity con- ditions on the integrand f. 6 Minimax Methods Step 3. Establish the sequential lower semicontinuity of G with respect to τ, i.e., Minimax methods are used to establish the exis- lim infn→∞ G(vn) ≥ G(v) whenever the sequence tence of saddle points of the functional (1), i.e., {vn} ⊂ X converges weakly to v ∈ X with re- critical points that are not extrema. More gener- spect to τ. 1 ally, for C functionals G : X → R where X is an Step 4. Conclude that u0 minimizes G. Indeed, infinite dimensional , as introduced in Section 5, the Palais-Smale compactness con- inf G(u) = lim G(un) = lim G(unk ) u∈X n→∞ k→∞ dition (P.-S.) plays the role of compactness in the ≥ G(u0) ≥ inf G(u). finite-dimensional case. Precisely, G satisfies the u∈X (P.-S.) condition if whenever {un} ⊂ X is such that {G(un)} is a bounded sequence in R and 7.2 Integrands: convex, polyconvex, 0 DG(un) → 0 in the dual of X, X , then {un} quasiconvex, rank-one convex admits a convergent subsequence. An important result for the existence of sad- In view of Step 3 above, it is important to charac- dle points that uses the (P.-S.) condition is the terize the class of integrands f in (1) for which the Mountain Pass Lemma of Ambrosetti and Rabi- corresponding functional F is sequentially lower nowitz, which states that if G satisfies the (P.-S.) semicontinuous with respect to τ. In the case 1,p d condition, if G(0) = 0 and there are r > 0 and in which X is the Sobolev space W (Ω; R ), 1 ≤ p ≤ +∞, and τ is the weak topology (weak- u0 ∈ X \ B(0, r) such that ? if p = +∞), this is related to convexity-type inf G > 0 and G(u0) ≤ 0, properties of f(x, u, ·). If min{d, N} = 1 then ∂B(0,r) under appropriate growth and regularity condi- then tions, it can be shown that convexity of f(x, u, ·) inf sup G(u) is necessary and sufficient. More generally, if γ∈C u∈γ min{d, N} > 1 then the corresponding condition is a critical value, where C is the set of all contin- is called quasiconvexity; precisely, f(x, u, ·) is said uous curves from [0, 1] into X joining 0 to u0. to be quasiconvex if In addition, minimax methods can be used to Z prove the existence of multiple critical points of f(x, u, ξ) ≤ fx, u, ξ + ∇ϕ(y) dy functionals G that satisfy certain symmetry prop- (0,1)N erties, for example, the generalization of the re- d×N 1,∞ N d sult by Ljusternik and Schnirelmann for symmet- for all ξ ∈ R and all ϕ ∈ W0 ((0, 1) ; R ), ric functions to the infinite dimensional case. whenever the right-hand side in this inequality is 5 well defined. Since this condition is nonlocal, in Ω ⊂ RN . Assume that the total mass of the applications in mechanics often one studies re- fluid is m, so that admissible density distributions R lated classes of integrands, such as polyconvex u :Ω → R satisfy the constraint Ω u(x) dx = m. and rank-one convex functions, for which there The total energy is given by the functional u 7→ R are algebraic criteria. Ω W (u(x)) dx, where W : R → [0, ∞) is the en- ergy per unit volume. Assume that W supports 8 Relaxation two phases a < b, that is, W is a double-well po- tential, with {u ∈ R : W (u) = 0} = {a, b}. Then In most applications Step 3 in Subsection 7.1 fails, any density distribution u that renders the body and this leads to an important topic at the core stable in the sense of Gibbs is a minimizer of the of the calculus of variations, namely the intro- following problem (P0) duction of a relaxed, or effective, energy G that Z Z  is related to G, as introduced in Section 5, as fol- min W (u(x)) dx : u(x) dx = m . (9) lows: Ω Ω (a) G is sequentially lower semicontinuous with If LN (Ω) = 1 and a < m < b, then given any respect to τ; N b−m measurable set E ⊂ Ω with L (E) = b−a , the function u = aχ +bχ is a solution of problem (b) G ≤ G and G inherits coercivity properties E Ω\E (9). This lack of uniqueness is due the fact that from G; interfaces between the two phases a and b are not (c) min G = inf G. penalized by the total energy. The physically pre- u∈X u∈X ferred solutions should be the ones that arise as When G is of the type (1), a central problem is limiting cases of a theory that penalizes interfa- to understand if G has an integral form of the cial energy, so it is expected that these solutions type (1) for some new integrand h, and if so then should minimize the surface area of ∂E ∩ Ω. what is the relation between h and the original In the van der Walls–Cahn–Hilliard theory of integrand f. phase transitions, the energy depends not only on If X = W 1,p(Ω), p ≥ 1, and τ is the weak topol- the density u but also on its gradient, precisely, ogy, then under appropriate growth and regular- Z Z ity conditions it can be shown that h(x, u, ·) is W (u(x)) dx + ε2 |∇u(x)|2 dx. the convex envelope of f(x, u, ·), i.e., the greatest Ω Ω convex function less than f(x, u, ·). In the vec- Note that the gradient term penalizes rapid torial case d > 1 the convex envelope is replaced changes of the density u, and thus it plays the by a similar notion of quasiconvex envelope (see role of an interfacial energy. Stable density dis- Subsection 7.2). tributions u are now solutions of the minimization problem (Pε) 9 Γ-Convergence Z Z  2 2 Often in physical problems the behavior of a sys- min W (u(x)) dx + ε |∇u(x)| dx , Ω Ω tem is described in terms of a sequence {Gn}, n ∈ N, of energy functionals Gn : X → [−∞, +∞] where the minimum is taken over all smooth func- R where X is a space with a metric d. Is tions u satisfying Ω u(x) dx = m. In 1983 Gurtin it possible to identify a limiting energy G∞ that conjectured that the limits, as ε → 0, of solutions captures qualitative properties of this family, and of (Pε) are solutions of (P0) with such that minimizers of Gn converge to minimiz- area. Using results of Modica and Mortola, this ers of G∞? conjecture was proved independently by Modica The notion of Γ-convergence, introduced by De and by Sternberg in the setting of Γ-convergence. Giorgi, provides a tool for answering these ques- The Γ-limit G∞ : X → [−∞, +∞] of {Gn} tions. To motivate this concept with an exam- with respect to a metric d, when it exists, is de- ple, consider a fluid confined into a container fined uniquely by the following properties: 6

(i) (liminf inequality) for every sequence {un} ⊂ 11 Symmetrization X converging to u ∈ X with respect to d Rearrangements of sets preserve their measure G∞(u) ≤ lim inf Gn(un); n→∞ while modifying their geometry to achieve spe- cific symmetries. In turn, rearrangements of a (ii) (limsup inequality) for every u ∈ X there function u yield new functions with desired sym- exists a sequence {un} ⊂ X converging to metry properties, and which are obtained via suit- u ∈ X with respect to d such that able rearrangements of the t-superlevel sets of u,Ωt := {x ∈ Ω: u(x) > t}. These tools G∞(u) ≥ lim sup Gn(un). are used in a variety of contexts, from harmonic n→∞ analysis and PDEs to spectral theory of differ- ential operators. In the calculus of variations, This notion may be extended to the case in which they may be often found in the study of ex- the convergence of the sequences is taken with trema of functionals of type (1). Among the respect to some weak topology rather than the most common rearrangements, we mention the topology induced by the metric d. In this con- directional monotone decreasing rearrangement, text, we remark that when the sequence {G } n the star-shaped rearrangement, the directional reduces to a single energy functional {G}, under Steiner symmetrization, the Schwarz symmetriza- appropriate growth and coercivity assumptions, tion, the circular and spherical symmetrization, G coincides with the relaxed energy G, as dis- ∞ and the radial symmetrization. cussed in Section 8. Other important applications of Γ-convergence Of these, we highlight the Schwarz symmetriza- include the Ginzburg-Landau theory for super- tion, which is the most frequently used in the cal- culus of variations. If u is a nonnegative mea- conductivity (see Subsection 13.5), homogeniza- N tion of variational problems (see Subsection 13.3), surable function with compact in R , dimension reduction problems in elasticity (see then its Schwartz symmetric rearrangement is the Subsection 13.2), free-discontinuity problems in (unique) spherically symmetric and decreasing function u? such that for all t > 0 the t-superlevel image segmentation in computer vision (see Sub- ? section 13.4) and in fracture mechanics. sets of u and u have the same measure. When Ω = RN , it can be shown that u? pre- serves the Lp of u and the regularity of u up 10 Regularity to first order, that is, if u belongs to W 1,p(RN ) then so does u?, 1 ≤ p ≤ +∞. Moreover, by the Optimal regularity of minimizers and local min- ? imizers of the energy (1) in the vectorial case P´olya-Szeg¨oinequality, ||∇u ||p ≤ ||∇u||p, and we remark that for p = ∞ this is obtained us- d ≥ 2, and when X = W 1,p(Ω; Rd), 1 ≤ p ≤ +∞, is mostly open. In the scalar case d = 1 there is ing the Brunn-Minkowski inequality discussed in an extensive body of literature on the regularity Subsection 2.3. of weak solutions of the Euler Lagrange equation Another important inequality relating u and ? (5), stemming from a fundamental result of De u is the Riesz inequality, and the Faber-Krahn Giorgi, independently obtained by Nash, in the inequality compares eigenvalues of the Dirich- ? late 1950’s. For d ≥ 2, in general (local) mini- let problems in Ω and in Ω . Classical appli- mizers of (1) are not everywhere smooth. On the cations of rearrangements include the derivation other hand, and under suitable hypotheses on the of the sharp constant in the Sobolev-Gagliardo- 1,p N integrand f, it can be shown that partial regular- Nirenberg inequality in W (R ), 1 < p < N, as ity holds, i.e., if u is a local minimizer then there well as in the Young inequality and the Hardy- exists an open subset of Ω, Ω0, of full measure Littlewood-Sobolev inequality. 1,α d such that u ∈ C (Ω0; R ), for some α ∈ (0, 1). Finally, we remark that the first and most im- Sharp estimates on α and on the Hausdorff di- portant application of Steiner symmetrization is mension of the singular set Σu := Ω \ Ω0 are still the isoperimetric property of balls (see Dido’s unknown. problem in Section 1). 7

12 Duality Theory deformations of the body can be described by maps u :Ω → R3. If the body is homogeneous Duality theory associates to a minimization prob- then the total elastic energy corresponding to u lem (P) a maximization problem (P∗), called the is given by the functional dual problem, and studies the relation between these two. It has important applications in sev- Z F (u) := f(∇u(x)) dx, (10) eral disciplines, including economics and mechan- Ω ics, and different areas of mathematics, such as the calculus of variations, convex analysis, and where f is the stored-energy density of the mate- . rial. In order to prevent interpenetration of mat- The theory of dual problems is inspired by the ter, the deformations should be invertible and it notion of duality in convex analysis and by the should require an infinite amount of energy to vi- Fenchel transform f ? of a function f : RN → olate this property, i.e., [−∞, +∞], defined as f(ξ) → +∞ as detξ → 0+. (11) ? N N f (η) := sup{η · ξ − f(ξ): ξ ∈ R } for η ∈ R . Also, f needs to be frame indifferent, i.e., As an example, consider the minimization problem f(Rξ) = f(ξ) (12) Z  1,p 3×3 (P) inf f(∇u) dx : u ∈ W0 (Ω) , for all rotations R and all ξ ∈ R . Ω Under appropriate coercivity and convex-type with f : RN → R. If f satisfies appropriate conditions on f (see Subsection 7.2), and bound- growth and convexity conditions, then the dual ary conditions, it can be shown that F admits ? problem (P ) is given by a global minimizer u0. However, the regularity of u is still an open problem and so the Euler- Z 0 n ? q N Lagrange equation cannot be derived (see Section sup − f (v(x)) dx : v ∈ L (Ω; R ), Ω 3). In addition, the existence of local minimizers o div v = 0 in Ω , remains unsolved.

1 1 where p + q = 1. The latter problem may be sim- 13.2 Dimension Reduction pler to handle in specific situations, e.g., for non- parametric minimal surfaces and with f given as An important problem in elasticity is the deriva- in (2), where, due to lack of coercivity, (P) may tion of models for thin structures, such as mem- not admit a solution in X. branes, shells, plates, rods, beams, etc., from the three dimensional elasticity theory. The mathe- 13 Some Contemporary Applications matical rigorous analysis was initiated by Acerbi, Buttazzo and Percivale in the 90’s for rods, fol- There is a plethora of applications of the calculus lowed by the work of Le Dret and Raoult for mem- of variations. Classical ones include Hamiltoni- branes. Recent contributions by Friesecke, James ans and Lagrangians, the Hamilton-Jacobi equa- and M¨ullerallowed to handle the physical require- tion, conservation laws, Noether’s theorem, opti- ments (11) and (12). The main tool underlying mal control. Below we focus on a few contempo- these works is Γ-convergence (see Section 9). rary applications that are pushing the frontiers of To illustrate the deduction in the case of mem- the theory in novel directions. branes, consider a thin cylindrical elastic body of thickness 2ε > 0 occupying the reference config- 2 13.1 Elasticity uration Ωε := ω × (−ε, ε), with ω ⊂ R . Using the typical rescaling (x1, x2, x3) 7→ (y1, y2, y3) := Consider an elastic body that occupies a domain (x1, x2, x3/ε), the deformations u of Ωε now cor- 3 Ω ⊂ R in a given reference configuration. The respond to deformation v of the fixed domain Ω1, 8

through the formula v(y1, y2, y3) = u(x1, x2, x3). Rudin-Osher-Fatemi total variation model), in- Therefore painting (e.g., recolorization). 1 Z Z  ∂v ∂v 1 ∂v  The Mumford-Shah model provides a good ex- f(∇u) dx = f , , dy. ample of the use of calculus of variations to treat ε ∂y1 ∂y2 ε ∂y3 Ωε Ω1 free discontinuity problems. Let Ω be a rectangle The right-hand side of the previous equality yields in the plane, representing the locus of the image a family of functionals to which the theory of Γ- with grey levels given by a function g :Ω → [0, 1]. convergence is applied. We want to find an approximation of g that is smooth outside a set K of sharp contours related 13.3 Homogenization to the set of discontinuities of g. This leads to the minimization of the functional Homogenization theory is used to describe the Z macroscopic behavior of heterogeneous compos- 2 2 ite materials, which are characterized by having (|∇u| + α(u − g) ) dx + βlength(K ∩ Ω), Ω\K two or more finely mixed material components. Composite materials have important technologi- over all contour curves K and functions u ∈ cal and industrial applications as their effective C1(Ω \ K), and where the first term forces u to properties are often better than the correspond- not vary much outside K, the second term asks ing properties of the individual constituents. The that u stays close to the original grey level g, study of these materials falls within the so-called and the last term ensures that K has length as multiscale problems, with the two relevant scales short as possible. The existence of a minimizing here being the microscopic scale at the level of pair (u, K) was established by De Giorgi, Car- the heterogeneities, and the macroscopic scale de- riero and Leaci, with u in a class of functions scribing the resulting “homogeneous” material. larger than C1(Ω \ K), the so-called functions of Mathematically, the properties of composite ma- special . The full regularity of terials can be described in terms of PDEs with these solutions u and the structure of K remain fast oscillating coefficients, or energy functionals an open problem. depending on a small parameter ε. As an ex- ample, consider a material matrix A with corre- 13.5 Ginzburg-Landau Theory for sponding stored energy density fA, with periodi- Superconductivity cally distributed inclusions of another material B with stored energy density fB, whose periodicity In the 1950’s Ginzburg and Landau proposed a cell has side-length ε. Then the total energy of mathematical theory to study phase transition the composite is given by problems in superconductivity; there are similar Z formulations to address problems in superfluids, [(1 − χ(x/ε))fA(∇u) + χ(x/ε)fB(∇u)] dx, e.g. helium II, and in XY-magnetism. In its sim- Ω plest form, the Ginzburg-Landau functional re- where χ is the characteristic function of the locus duces to of material B contained in the unit cube Q of ma- 3 Z Z terial A, extended periodically to R with period 1 2 1 2 2 Fε(u) := |∇u| dx + 2 (|u| − 1) dx, Q. The goal here its to characterize the “homog- 2 Ω 4ε Ω enized” energy when ε → 0+ using Γ-convergence 2 (see Section 9). where Ω ⊂ R is a star-shaped domain, the con- densate wave function u ∈ W 1,2(Ω; R2) is an or- 13.4 Computer Vision der parameter with two degrees of freedom, and the parameter ε is a (small) characteristic length. Several problems in computer vision can be Given g : ∂Ω → S1, with S1 the unit circle in R2 treated variationally, including image segmen- centered at the origin, we are interested in charac- tation (e.g., the Mumford-Shah and the Blake- terizing the limits of minimizers uε of Fε subject Zisserman models), image morphing, image de- to the boundary condition uε = g on ∂Ω. Un- noising (e.g., the Perona-Malik scheme and the der suitable geometric conditions on g (related to 9

1 the winding number), Bethuel, Brezis and H´elein St(w0), t > 0, is the unique C solution of the have shown that there are no limiting functions Cauchy problem u in C(Ω, 1) satisfying the boundary condition. S d Rather, the limiting functions are smooth out- wt = −∇h(wt) for t > 0, lim wt = w0, side a finite set of singularities, called vortices. dt t→0+ (13) Γ-convergence techniques may be used to study if it exists. this family of functionals (see Section 9). If D2h ≥ αI for some α ∈ R, then it can be shown that the gradient flow exists, it is unique, 13.6 Mass Transport and it satisfies a semigroup property, i.e.,

Mass transportation was introduced by Monge in St+s(w0) = St(Ss(w0)), lim St(w0) = w0 + 1781, studied by the Nobel prize winner Kan- t→0 m torovich in the 1940’s and, and revived by Brenier for every w0 ∈ R . in 1987. Since then, it has surfaced in a variety of A common way used to approximate discretely areas, from economics to optimization. Given a the solution of (13) is the implicit Euler scheme: pile of sand of mass one and a hole of volume one, given a time step τ > 0, consider the partition of we want to fill the hole with the sand while min- [0, +∞) imizing the cost of transportation. This problem 0 1 n is formulated using probability theory as follows. {0 = tτ < tτ < ··· < tτ < ···}, The pile and the hole are represented by proba- where tn := nτ. Define recursively a discrete se- bility measures µ and ν, with supports in measur- τ quence {W n} as follows: assuming that W n−1 able spaces X and Y , respectively. If A ⊂ X and τ τ has already been defined, let W n be the unique B ⊂ Y are measurable sets, then µ(A) measures τ minimizer of the function the amount of sand in A, and ν(B) measures the 1 amount of sand that can fill B. The cost of trans- w 7→ |w − W n−1|2 + h(w). (14) portation is modeled by a measurable cost func- 2τ τ tion c : X ×Y → ∪+∞. Kantorovich’s optimal R Introduce the piecewise linear function Wτ : transportation problem consists of minimizing m [0, +∞) → R given by Z n−1 n c(x, y) dπ(x, y) t − tτ n−1 tτ − t n Wτ (t) = Wτ + Wτ X×Y τ τ n−1 n 0 + over all probability measures on X × Y such that for t ∈ [tτ , tτ ]. If Wτ → w0 as τ → 0 then π(A × Y ) = µ(A) and π(X × B) = ν(B), for it can be shown that {Wτ }τ>0 converges to the + all measurable sets A ⊂ X and B ⊂ Y . The solution of (13) as τ → 0 . main problem is to establish existence of mini- This approximation scheme, here described for d mizers and to obtain their characterization. This the finite dimensional vector space R , may be d depends strongly on the cost function c and on extended to the case in which R is replaced by the regularity of the measures µ and ν. There an infinite dimensional metric space X, the func- is a multitude of applications of this theory, and tion h is replaced by a functional G : X → R, here we mention that it can be used to give a sim- and the minimization procedure in (14) is now a ple proof of the Brunn-Minkowski inequality (see variational minimization problem of the type ad- (4)). dressed in Section 7, known as De Giorgi’s min- imizing movements. Important applications in- clude the study of a large class of parabolic PDEs. 13.7 Gradient Flows

Given a function h : Rm → R of class C2, the Further Reading m gradient flow of h is the family of maps St : R → 1. Ambrosio, L., Fusco, N., and Pallara, D. 2000. m R , t ≥ 0, satisfying the following property: for Functions of Bounded Variation and Free Dis- m every w0 ∈ R , S0(w0) := w0 and the curve wt := continuity Problems. Oxford: Clarendon Press. 10

2. Ambrosio, L., Gigli, N., and Savar´e, G. 2008. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: Birkh¨auser. 3. Braides, A. 2002. Γ-Convergence for Beginners. New York: Oxford University Press. 4. Bethuel, F., Brezis, H. and H´elein, F. 1994. Ginzburg-Landau Vortices. Boston: Birkh¨auser. 5. Dacorogna, B. 2008. Direct Methods in the Cal- culus of Variations. New York: Springer. 6. Dal Maso, G. 1993. An Introduction to Γ- Convergence. Boston: Birkh¨auser. 7. Cioranescu, D. and Donato, P. 1999. An Intro- duction to Homogenization. New York: Oxford University Press. 8. Ekeland, I. and T´eman,R. 1999. Convex Anal- ysis and Variational Problems. Philadelphia: SIAM. 9. Fonseca, I. and Leoni, G. 2007. Modern Methods in the Calculus of Variations: Lp Spaces. New York: Springer. 10. Giusti, E. 1984. Minimal Surfaces and Func- tions of Bounded Variation. Boston - Basel - Stuttgart: Birk¨auser. 11. Giusti, E. 2003. Direct Methods in the Calculus of Variation. Singapore: World Scientific. 12. Kawohl, B. 1985. Rearrangements and Con- vexity of Level Sets in PDE. Berlin: Springer- Verlag. 13. Kinderlehrer, D. and Stampacchia, G. 2000. an Introduction to Variational Inequalities and Their Applications. Philadelphia: SIAM. 14. Rabinowitz, P. H. 1986. Minimax Methods in Critical Point Theory with Applications to Dif- ferential Equations. Providence: Amer. Math. Soc. 15. Rockafellar, R. T. 1997. Convex Analysis. Prin- centon: Princeton University Press. 16. Struwe, M. 2008. Variational methods. Applica- tions to nonlinear partial differential equations and Hamiltonian systems. Berlin: Springer- Verlag. 17. Villani, C., 2003. Topics in Optimal Transporta- tion. Providence: Amer. Math. Soc.