Calculus of Variations we consider a functional Z I. Fonseca and G. Leoni u 2 X 7! F (u) := f(x; u(x); ru(x)) dx; (1) Carnegie Mellon University, USA Ω where X is a function space (usually a Lp space or a Sobolev-type space), u :Ω ! Rd, with Ω ⊂ RN 1 History an open set, N and d are positive integers, and the density is a function f(x; u; ξ), with (x; u; ξ) 2 The calculus of variations is a branch of mathe- Ω × Rd × Rd×N . Here, and in what follows, ru 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 2 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 jγ (ti) − γ (ti−1)j ; 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 2 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 jγ (t)j dt: finitesimal calculus and was later found by Jacob In the absence of constraints, the geodesic is the Bernoulli, Newton, Leibniz and de l'H^opital.The straight segment with endpoints A and B, and so arguments thus developed led to the development L(γ) = jA − Bj. Often in applications the curves of the foundations of the calculus of variations by are restricted to lie on a given manifold, e.g., a Euler. Important contributions to the subject are sphere (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 surface 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 integral 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 measure 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 spheres, 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 open set Ω ⊂ R and a smooth func- Wulff set tion u :Ω ! R, then the area of the graph of u, f(x; u(x)) : x 2 Ωg, 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 + jruj2 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 ru a given volume. Precisely, we seek to minimize div = 0 in Ω: p1 + jruj2 the surface integral Z (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 Gaussian curvature 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 ≤ +1, of all Ω, then the variation u+t' is admissible if ' ≥ 0 functions u 2 Lp(Ω) whose distributional gradi- and t ≥ 0. Therefore, the function g satisfies ent ru belongs to Lp(Ω; RN ). Let u 2 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 @' f(x; u(x); ru(x)) dx ≤ f(x; v(x); rv(x)) dx (x; u; ru) U U @ξ @x Ω i=1 i i for every open subset Ucompactly contained in @f 1;p + (x; u; ru)' dx ≥ 0 Ω, and all v such that u − v 2 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 ' 2 Cc (Ω). This is called the v will then coincide with u outside the set U. If obstacle problem, and the coincidence set fu = φg 1 is not known a priori and is called the free bound- ' 2 Cc (Ω) then u + t', t 2 R, are admissible, and thus ary. This is an example of a broad class of vari- ational inequalities and free boundary problems t 2 R 7! g(t) := F (u + t') that have applications in a variety of contexts, including the modeling of the melting of ice (the has a minimum at t = 0.