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calculus of variations we consider a functional i fonseca and g leoni u x7 f u z f x u x u x dx 1 carnegie mellon university usa wherex ...

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                   Calculus of Variations                                   we consider a functional
                   I. Fonseca and G. Leoni                                    u∈X7→F(u):=Z f(x,u(x),∇u(x))dx, (1)
                   Carnegie Mellon University, USA                                                 Ω
                                                                            whereX isafunctionspace(usuallyaLp spaceor
                                                                            a Sobolev-type space), u : Ω → Rd, with Ω ⊂ RN
                   1 History                                                an open set, N and d are positive integers, and
                   The calculus of variations is a branch of mathe-         the density is a function f(x,u,ξ), with (x,u,ξ) ∈
                   matical analysis that studies extrema and critical       Ω×Rd×Rd×N. Here, and in what follows, ∇u
                   points of functionals (or energies). Here, by func-      stands for the d×N matrix-valued distributional
                   tional we mean a mapping from a function space           derivative of u.
                   to the real numbers.                                       The calculus of variations is a vast theory and
                                                                            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            2 Extrema
                   who emigrated to North Africa and upon arrival
                   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          ampleistheproblemoffindingthecurveofshort-
                   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 n                     o
                   plane but on different vertical lines. Assume that              L(γ) := sup    X|γ(ti)−γ(ti−1)| ,
                   Aishigher than B, and that a particle M is mov-
                   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-         [0,1]. If γ is smooth, then L(γ) = R1|γ′(t)| dt.
                   finitesimal calculus and was later found by Jacob                                                 0
                                                                            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 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.
                     ThecommonfeatureunderlyingDido’sandthe                 2.1    Minimal Surfaces
                   brachistochrone problems is that one seeks to
                   maximize or minimize a functional over a class           Aminimalsurface isasurfaceofleastareaamong
                   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,     The understanding of the surface structure of
                 {(x,u(x)) : x ∈ Ω}, is given by                      crystals plays a central role in many fields of
                                     Z p          2                   physics, chemistry and materials science. If the
                            F(u) := Ω     1+|∇u| dx.           (2)    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
                          div   p ∇u          =0 inΩ.                 a given volume. Precisely, we seek to minimize
                                           2                          the surface integral
                                  1+|∇u|                                               Z
                 2.2    Willmore Functional                                                ψ(ν(x))dσ                (3)
                                                                                        ∂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                  k +k
                 W(S) :=       H dσ, where H := 1 2 and k1            shape. AkeyingredientintheproofistheBrunn-
                             S                        2
                 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)) (4)
                 ing energy), to mathematical biology (e.g., cell
                 membranes) to image segmentation in computer         which holds for all Lebesgue measurable sets
                 vision (e.g., staircasing).                          A,B ⊂RN suchthatA+B isalsoLebesguemea-
                                                                                       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.
                             ∆ H+2H(H2−K)=0,                          3 The Euler Lagrange Equation
                               S
                 where K := k1k2 is the Gaussian curvature and        Consider the functional (1), in the scalar case
                 ∆ is the Laplace-Beltrami operator.                  d = 1, and where f of class C1 and X is the
                   S
                                                                                                                        3
                  Sobolev space X = W1,p(Ω), 1 ≤ p ≤ +∞, of all         Ω, then the variation u+tϕ is admissible if ϕ ≥ 0
                                  p
                  functions u ∈ L (Ω) whose distributional gradi-       and t ≥ 0. Therefore, the function g satisfies
                                       p      N                          ′
                  ent ∇u belongs to L (Ω;R ). Let u ∈ X be a            g (0) ≥ 0, and the Euler-Lagrange equation (5)
                  local minimizer of the functional F, that is,         becomes the variational inequality
                  Z                         Z                             Z  N
                   U f(x,u(x),∇u(x))dx ≤ U f(x,v(x),∇v(x))dx                    X∂f(x,u,∇u)∂ϕ
                                                                                    ∂ξ           ∂x
                                                                            Ω   i=1   i             i
                  for every open subset Ucompactly contained in                                 ∂f            
                  Ω, and all v such that u − v ∈ W1,p(U), where                              +     (x,u,∇u)ϕ dx≥0
                    1,p                              0                                          ∂u
                  W (U)is the space of all functions in W1,p(U)
                    0                                                   for all nonnegative ϕ ∈ C1(Ω). This is called the
                  “vanishing” on the boundary of ∂U. Note that                                   c
                  v will then coincide with u outside the set U. If     obstacle problem, and the coincidence set {u = φ}
                  ϕ ∈ C1(Ω) then u + tϕ, t ∈ R, are admissible,         is not known a priori and is called the free bound-
                         c
                  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.
                  g′(0) = 0, i.e.,                                        A related class of minimization problems in
                     Z  N                                              which the unknowns are both an underlying field
                          X∂f(x,u,∇u)∂ϕ                                 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
                                   +∂u(x,u,∇u)ϕ dx=0. (5)               (1) and a surface energy, e.g., as in (3). Impor-
                                                                        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
                                               ∂f                       5 Lagrange Multipliers
                         div(∇ f(x,u,∇u)) =       (x,u,∇u),     (6)
                               ξ               ∂u                       The method of Lagrange multipliers in Banach
                  where ∇ f(x,u,ξ) is the gradient of the function      spaces is used to find extrema of functionals G :
                          ξ                                             X→Rsubject to a constraint
                  f(x,u,·).
                    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
                                                                        if G and Ψ are of class C1 and u ∈ X is an ex-
                  4 Variational Inequalities, Free                      tremum of G subject to (7), and if the derivative
                      Boundary and Free Discontinuity                   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 ∈ W1,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                onafunctionalG, asintroducedinSection5, that
                    amongallcritical points of the family of function-            guarantee the existence of minimizers of G. The
                    als G+λΨ, λ ∈ R.                             1,p      d       method consists of the following steps:
                       If G has the form (1) and X = W              (Ω;R ),       Step 1. Consider a minimizing sequence {u } ⊂
                    1 ≤ p ≤ +∞, then typical examples of Ψ are                                                                       n
                                                                                  X, i.e., lim       G(u ) = inf        G(u).
                              Z                                 Z                              n→∞        n        u∈X
                                                                                  Step 2. Prove that {u } admits a subsequence
                                     s                                                                        n
                    Ψ(u) :=       |u| dx−c1      or   Ψ(u) :=      udx−c2         {u } converging to some u ∈ X with respect
                                                                                     n                              0
                                Ω                                Ω                     k
                                                                                  to some (weak) topology τ in X. When G has
                    for some constants c1 ∈ R, c2 ∈ Rd, 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-               liminfn→∞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.
                    ally, for C1 functionals G : X → R where X is an              Step 4. Conclude that u minimizes G. Indeed,
                                                                                                                0
                    infinite dimensional Banach space, as introduced
                                                                                         inf G(u) = lim G(u ) = lim G(u )
                    in Section 5, the Palais-Smale compactness con-                     u∈X           n→∞        n     k→∞        nk
                    dition (P.-S.) plays the role of compactness in the                             ≥G(u )≥ inf G(u).
                    finite-dimensional case. Precisely, G satisfies the                                      0     u∈X
                    (P.-S.) condition if whenever {u } ⊂ X is such
                                                          n
                    that {G(u )} is a bounded sequence in R and                   7.2     Integrands: convex, polyconvex,
                                 n
                    DG(u ) → 0 in the dual of X, X′, then {u }
                           n                                              n               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
                    condition, if G(0) = 0 and there are r > 0 and                in which X is the Sobolev space W1,p(Ω;Rd),
                    u ∈X\B(0,r) such that                                         1 ≤ p ≤ +∞, and τ is the weak topology (weak-
                      0                                                           ⋆ if p = +∞), this is related to convexity-type
                                inf   G>0 and G(u )≤0,
                                                           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 supG(u)                             is necessary and sufficient.         More generally, if
                                          γ∈Cu∈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 u .
                                                                        0         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) ≤          N fx,u,ξ +∇ϕ(y)dy
                    functionals G that satisfy certain symmetry prop-                                  (0,1)
                    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 ∈ W        ((0,1) ;R ),
                                                                                                                        0
                    ric functions to the infinite dimensional case.                whenever the right-hand side in this inequality is
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...Calculus of variations we consider a functional i fonseca and g leoni u x f z dx carnegie mellon university usa wherex isafunctionspace usuallyalp spaceor sobolev type space rd with rn history an open set n d are positive integers the is branch mathe density function matical analysis that studies extrema critical here in what follows points functionals or energies by func stands for matrix valued distributional tional mean mapping from derivative to real numbers vast theory chose highlight some contemporary as one rst questions may be framed pects eld conclude this article within dido s isoperimetric prob mentioning few forefront areas application lem see subsection nd shape driving current research curve prescribed perimeter maximizes area enclosed was phoenician princess who emigrated north africa upon arrival obtained native chief much territory section address fundamental minimiza she could enclose ox hide cut tion problems relevant techniques cal into long strip used it delineate ...

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