Filetype PDF | Posted on 25 Jan 2023 | 2 years ago
Authentication
digital resources
158x Filetype PDF File size 0.29 MB Source: cvgmt.sns.it
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 f x,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
no reviews yet
Please Login to review.
