Ebook Partial differential equations in action Part 2

(BQ) Part 2 book Partial differential equations in action has contents: Distributions and sobolev spaces, elements of functional analysis, variational formulation of elliptic problems, elements of functional analysis.

Elements of Functional Analysis

Motivations – Norms and Banach Spaces – Hilbert Spaces – Projections and Bases – ear Operators and Duality – Abstract Variational Problems – Compactness and WeakConvergence – The Fredholm Alternative – Spectral Theory for Symmetric BilinearForms

Lin-6.1 Motivations

The main purpose in the previous chapters has been to introduce part of the basicand classical theory of some important equations of mathematical physics Theemphasis on phenomenological aspects and the connection with a probabilisticpoint of view should have conveyed to the reader some intuition and feeling aboutthe interpretation and the limits of those models

The few rigorous theorems and proofs we have presented had the role of ing to light the main results on the qualitative properties of the solutions andjustifying, partially at least, the well-posedness of the relevant boundary and ini-tial/boundary value problems we have considered

bring-However, these purposes are somehow in competition with one of the mostimportant role of modern mathematics, which is to reach a unifying vision of largeclasses of problems under a common structure, capable not only of increasingtheoretical understanding, but also of providing the necessary flexibility to guidethe numerical methods which will be used to compute approximate solutions.This conceptual jump requires a change of perspective, based on the introduc-tion of abstract methods, historically originating from the vain attempts to solvebasic problems (e.g in electrostatics) at the end of the 19th century It turns outthat the new level of knowledge opens the door to the solution of complex problems

6.1 Motivations 303

It could be useful for understanding the subsequent development of the theory,

to examine in an informal way how the main ideas come out, working on a couple

of specific examples

Let us go back to the derivation of the diffusion equation, in subsection 2.1.2 Ifthe body is heterogeneous or anisotropic, may be with discontinuities in its thermalparameters (e.g due to the mixture of two different materials), the Fourier law ofheat conduction gives for the flux function q the form

q =−A (x) ∇u,where the matrix A satisfies the condition

q·∇u = −A (x) ∇u · ∇u ≤ 0 (ellipticity condition),

reflecting the tendency of heat to flow from hotter to cooler regions If ρ = ρ (x)and cv= cv(x) are the density and the specific heat of the material, and f = f (x)

is the rate of external heat supply per unit volume, we are led to the diffusionequation

ρcvut− div (A (x) ∇u) = f

In stationary conditions, u (x,t) = u (x), and we are reduced to

−div (A (x) ∇u) = f (6.1)Since the matrix A encodes the conductivity properties of the medium, we expect

a low degree of regularity of A, but then a natural question arises: what is themeaning of equation (6.1) if we cannot compute the divergence of A?

We have already faced similar situations in subsections 4.4.2, where we haveintroduced discontinuous solutions of a conservation law, and in subsection 5.4.2,where we have considered solutions of the wave equation with irregular initial data.Let us follow the same ideas

Suppose we want to solve equation (6.1) in a bounded domain Ω, with zeroboundary data (Dirichlet problem) Formally, we multiply the differential equation

by a smooth test function vanishing on ∂Ω, and we integrate over Ω:

which is called weak or variational formulation of our Dirichlet problem

Equation (6.2) makes perfect sense for A and f bounded (possibly ous) and u, v∈ ˚C1

discontinu-Ω, the set of of functions in C1

Ω, vanishing on ∂Ω Then,

we may say that u ∈ ˚C1

Ω

is a weak solution of our Dirichlet problem if (6.2)

304 6 Elements of Functional Analysis

holds for every v∈ ˚C1

Ω Fine, but now we have to prove the well-posedness ofthe problem so formulated!

Things are not so straightforward, as we have experienced in section 4.4.3 and,actually, it turns out that ˚C1

Ω

is not the proper choice, although it seems to

be the natural one To see why, let us consider another example, somewhat morerevealing

Consider the equilibrium position of a stretched membrane having the shape

of a square Ω, subject to an external load f (force per unit mass) and kept at levelzero on ∂Ω

Since there is no time evolution, the position of the membrane may be described

by a function u = u (x), solution of the Dirichlet problem

Ωfv expresses the work done by theexternal forces

Thus, the weak formulation (6.4) states that these two works balance, whichconstitutes a version of the principle of virtual work

There is more, if we bring into play the energy In fact, the total potential energy

is proportional to

E (v) =

Ω

|∇v|2dx /0 1

internal elastic energy

−

Ω

fv dx /0 1

Thus, changing point of view, instead of looking for a weak solution of (6.4)

we may, equivalently, look for a minimizer of (6.5)

However there is a drawback It turns out that the minimum problem does nothave a solution, except for some trivial cases The reason is that we are looking inthe wrong set of admissible functions

of the derivatives, actually neither of u The space ˚C1

Ω

is too narrow to haveany hope of finding the minimizer there Thus, we are forced to enlarge the set ofadmissible functions and the correct one turns out to be the so called Sobolev space

H1(Ω), whose elements are exactly the functions belonging to L2(Ω), togetherwith their first derivatives, vanishing on ∂Ω We could call them functions of finiteenergy!

Although we feel we are on the right track, there is a price to pay, to puteverything in a rigorous perspective and avoid risks of contradiction or non-senses

In fact many questions arise immediately

For instance, what do we mean by the gradient of a function which is only

in L2(Ω), maybe with a lot of discontinuities? More: a function in L2(Ω) is, inprinciple, well defined except on sets of measure zero But, then, what does it mean

“vanishing on ∂Ω”, which is precisely a set of measure zero?

We shall answer these questions in Chapter 7 We may anticipate that, for thefirst one, the idea is the same we used to define the Dirac delta as a derivative

of the Heaviside function, resorting to a weaker notion of derivative (we shall say

in the sense of distributions), based on the miraculous formula of Gauss and theintroduction of a suitable set of test function

For the second question, there is a way to introduce in a suitable coherent way

a so called trace operator which associates to a function u∈ L2(Ω), with gradient

in L2(Ω), a function u|∂Ω representing its values on ∂Ω (see subsection 6.6.1).The elements of H1(Ω) vanish on ∂Ω in the sense that they have zero trace.Another question is what makes the space H1(Ω) so special Here the con-junction between geometrical and analytical aspects comes into play First of all,although it is an infinite-dimensional vector space, we may endow H1

0(Ω) with astructure which reflects as much as possible the structure of a finite dimensionalvector space likeRn, where life is obviously easier

Indeed, in this vector space (thinking ofR as the scalar field) we may introduce

an inner product given by

(u, v)1= 0

Having defined the inner product (·, ·)1, we may define the size (norm) of u by

u1=

(u, u)1

306 6 Elements of Functional Analysis

and the distance between u and v by

dist (u, v) =u − v1.Thus, we may say that a sequence {un} ⊂ H1(Ω) converges to u in H1(Ω) if

dist (un, u)→ 0 as n→ ∞

It may be observed that all of this can be done, even more comfortably, in thespace ˚C1

Ω

This is true, but with a key difference

Let us use an analogy with an elementary fact The minimizer of the function

f (x) = (x− π)2

does not exist among the rational numbersQ, although it can be approximated asmuch as one likes by these numbers If from a very practical point of view, rationalnumbers could be considered satisfactory enough, certainly it is not so from thepoint of view of the development of science and technology, since, for instance,

no one could even conceive the achievements of Calculus without the real numbersystem

As R is the completion of Q, in the sense that R contains all the limits ofsequences inQ that converge somewhere, the same is true for H1(Ω) with respect

In fact, (6.4) means that we are searching for an element u, whose inner productwith any element v of H1(Ω) reproduces “the action of f on v”, given by the linearmap

F (ax + by) = aF (x) + bF (y) ∀a, b ∈ R, ∀x, y ∈Rn,

can be expressed as the inner product with a unique representative vector zF∈Rn

(Representation Theorem) This amounts to saying that there is exactly one tion zF of the equation

solu-z· y = F (y) for every y∈Rn (6.6)The structure of the two equations (6.4), (6.6) is the same: on the left hand sidethere is an inner product and on the other one a linear map

Another natural question arises: is there any analogue of the RepresentationTheorem in H1(Ω)?

The answer is yes (see Riesz’s Theorem 6.3), with a little effort due to theinfinite dimension of H01(Ω) The Hilbert space structure of H01(Ω) plays a key

6.2 Norms and Banach Spaces 307

role This requires the study of linear functionals and the related concept of dualspace Then, an abstract result of geometric nature, implies the well-posedness of

a concrete boundary value problem

What about equation (6.2)? Well, if the matrix A is symmetric and strictlypositive, the left hand side of (6.2) still defines an inner product in H1(Ω) andagain Riesz’s Theorem yields the well-posedness of the Dirichlet problem

If A is not symmetric, things change only a little Various generalizations ofRiesz’s Theorem (e.g the Lax-Milgram Theorem 6.4) allow the unified treatment

of more general problems, through their weak or variational formulation Actually,

as we have experienced with equation (6.2), the variational formulation is often theonly way of formulating and solving a problem, without losing its original features.The above arguments should have convinced the reader of the existence of ageneral Hilbert space structure underlying a large class of problems, arising in theapplications In this chapter we develop the tools of Functional Analysis, essentialfor a correct variational formulation of a wide variety of boundary value problems.The results we present constitute the theoretical basis for numerical methods such

as finite elements or more generally, Galerkin’s methods, and this makes the theoryeven more attractive and important

More advanced results, related to general solvability questions and the spectralproperties of elliptic operators are included at the end of this chapter

A final comment is in order Look again at the minimization problem above Wehave enlarged the class of admissible configurations from a class of quite smoothfunctions to a rather wide class of functions What kind of solutions are we find-ing with these abstract methods? If the data (e.g Ω and f, for the membrane)are regular, could the corresponding solutions be irregular? If yes, this does notsound too good! In fact, although we are working in a setting of possibly irregularconfigurations, it turns out that the solution actually possesses its natural degree

of regularity, once more confirming the intrinsic coherence of the method

It also turns out that the knowledge of the optimal regularity of the solutionplays an important role in the error control for numerical methods However, thispart of the theory is rather technical and we do not have much space to treat it

in detail We shall only state some of the most common results

The power of abstract methods is not restricted to stationary problems As weshall see, Sobolev spaces depending on time can be introduced for the treatment

of evolution problems, both of diffusive or wave propagation type (see Chapter 7).Also, in this introductory book, the emphasis is mainly to linear problems

6.2 Norms and Banach Spaces

It may be useful for future developments, to introduce norm and distance pendently of an inner product, to emphasize better their axiomatic properties.Let X be a linear space over the scalar field R or C A norm in X, is a realfunction

308 6 Elements of Functional Analysis

such that, for each scalar λ and every x,y∈ X, the following properties hold:

1 x ≥ 0; x = 0 if and only if x = 0 (positivity)

2 λx = |λ| x (homogeneity)

3 x + y ≤ x + y (triangular inequality)

A norm is introduced to measure the size (or the “length”) of each vector x∈ X,

so that properties 1, 2, 3 should appear as natural requirements

A normed space is a linear space X endowed with a norm · With a norm isassociated the distance between two vectors given by

d (x, y) =x − y

which makes X a metric space and allows to define a topology in X and a notion

of convergence in a very simple way

We say that a sequence {xn} ⊂ X converges to x in X, and we write xm→ x

6.2 Norms and Banach Spaces 309

Let X, Y linear spaces, endowed with the norms·X and·Y, respectively,and let F : X→ Y We say that F is continuous at x ∈ X if

F (y) − F (x)Y → 0 when y − xX→ 0

or, equivalently, if, for every sequence{xm} ⊂ X,

xm− xX→ 0 implies F (xm)− F (x)Y → 0

F is continuous in X if it is continuous at every x∈ X In particular:

Proposition 6.1 Every norm in a linear space X is continuous in X

Proof Let· be a norm in X From the triangular inequality, we may write

y ≤ y − x + x and x ≤ y − x + y

whence

|y − x| ≤ y − x Thus, if y − x → 0 then |y − x| → 0, which is the continuity of the norm



Some examples are in order

Spaces of continuous functions Let X = C (A) be the set of (real orcomplex) continuous functions on A, where A is a compact subset ofRn, endowedwith the norm (called maximum norm)

Note that other norms may be introduced in C (A), for instance the leastsquares or L2(A) norm

fL2 (A)=

A

|f|2

1/2

Equipped with this norm C (A) is not complete Let, for example A = [−1, 1] ⊂ R.The sequence

310 6 Elements of Functional Analysis

contained in C ([−1, 1]), is a Cauchy sequence with respect to the L2norm In fact(letting m > k),

1

m+

1k

To denote a derivative of order m, it is convenient to introduce an n− uple ofnonnegative integers, α = (α1, , αn), called multi-index, of length

|α| = α1+ + αn = m,and set

Dα= ∂

α1

∂xα1 1

∂

α n

∂xαn n

of sequences, it follows that the resulting space is a Banach space

Remark 6.1 With the introduction of function spaces we are actually making astep towards abstraction, regarding a function from a different perspective Incalculus we see it as a point map while here we have to consider it as a singleelement (or a point or a vector) of a vector space

Summable and bounded functions Let Ω be an open set inRn and p≥ 1

a real number Let X = Lp(Ω) be the set of functions f such that|f|p is Lebesgueintegrable in Ω Identifying two functions f and g when they are equal a.e.1in Ω,

1 A property is valid almost everywhere in a set Ω, a.e in short, if it is true at all points

in Ω, but for a subset of measure zero (Appendix B)

of Lp(Ω) is not a single function but, actually, an equivalence class of functions,different from one another only on subsets of measure zero At first glance, thisfact could be annoying, but after all, the situation is perfectly analogous to con-sidering a rational number as an equivalent class of fractions (2/3, 4/6, 8/12 represent the same number) For practical purposes one may always refer to themore convenient representative of the class.

Let X = L∞(Ω) the set of essentially bounded functions in Ω Recall3 that

f : Ω→ R (or C) is essentially bounded if there exists M such that

|f (x)| ≤ M a.e in Ω (6.9)The infimum of all numbers M with the property (6.9) is called essential supremum

of f, and denoted by

fL∞ (Ω)= ess sup

Ω |f|

If we identify two functions when they are equal a.e., fL∞ (Ω) is a norm in

L∞(Ω), and L∞(Ω) becomes a Banach space

H¨older inequality (1.9) mentioned in chapter 1, may be now rewritten in terms

312 6 Elements of Functional Analysis

with the following three properties For every x, y, z∈ X and scalars λ, μ ∈ R:

1 (x, x)≥ 0 and (x, x) = 0 if and only if x = 0 (positivity)

2 (x, y) = (y, x) (symmetry)

3 (μx + λy, z) = μ (x, z) + λ (y, z) (bilinearity)

A linear space endowed with an inner product is called an inner product space.Property 3 shows that the inner product is linear with respect to its first argument.From 2, the same is true for the second argument as well Then, we say that (·, ·)constitutes a symmetric bilinear form in X When different inner product spacesare involved it may be necessary the use of notations like (·, ·)X, to avoid confusion.Remark 6.2 If the scalar field isC, then

(·, ·) : X × X → Cand property 2 has to be replaced by

2bis (x, y) = (y, x) where the bar denotes complex conjugation As a consequence,

we have

(z, μx + λy) = μ (z, x) + λ (z, y)and we say that (·, ·) is antilinear with respect to its second argument or that it

(1) Schwarz’s inequality:

|(x, y)| ≤ x y (6.12)Moreover equality holds in (6.12) if and only if x and y are linearly dependent.(2) Parallelogram law:

x + y2+x − y2= 2x2+ 2y2.The parallelogram law generalizes an elementary result in euclidean plane ge-ometry: in a parallelogram, the sum of the squares of the sides length equals thesum of the squares of the diagonals length The Schwarz inequality implies thatthe inner product is continuous; in fact, writing

(w, z)− (x, y) = (w − x, z) + (x, z − y)

6.3 Hilbert Spaces 313

we have

|(w, z) − (x, y)| ≤ w − x z + x z − y

so that, if w→ x and z → y, then (w, z) → (x, y)

Proof (1) We mimic the finite dimensional proof Let t ∈ R and x, y ∈ X.Using the properties of the inner product and (6.11), we may write:

0≤ (tx + y, tx + y) = t2x2+ 2t (x, y) +y2≡ P (t)

Thus, the second degree polynomial P (t) is always nonnegative, whence

(x, y)2− x2y2≤ 0which is the Schwarz inequality Equality is possible only if tx + y = 0, i.e if x and

y are linearly dependent

(2) Just observe that

x ± y2= (x± y, y ± y) = x2± 2 (x, y) + y2 (6.13)



Definition 6.2 Let H be an inner product space We say that H is a Hilbertspace if it is complete with respect to the norm (6.11), induced by the innerproduct

Two Hilbert spaces H1 and H2 are isomorphic if there exists a linear map

L : H1→ H2 which preserves the inner product, i.e.:

(x, y)H1 = (Lx, Ly)H2 ∀x, y ∈ H1

In particular

xH1 =LxH2Example 6.1.Rnis a Hilbert space with respect to the usual inner product

314 6 Elements of Functional Analysis

defines another scalar product inRn Actually, every inner product inRn may bewritten in the form (6.14), with a suitable matrix A

Cn is a Hilbert space with respect to the inner product

If Ω is fixed, we will simply use the notations (u, v)0 instead of (u, v)L2 (Ω)

andu0instead ofuL2 (Ω)

C a Hilbert space over C (seeProblem 6.3) This space constitutes the discrete analogue of L2(0, 2π) Indeed,each u∈ L2(0, 2π) has an expansion in Fourier series (Appendix A)

u(x) =

m∈Z

?umeimx,where

?um= 12π

2π 0

u (x) e−imxdx

Note that ?um=?u−m, since u is a real function From Parseval’s identity, we have

(u, v)0=

2π 0

u2= 2π

m∈Z

|?um|2

6.3 Hilbert Spaces 315

Example 6.4 A Sobolev space It is possible to use the frequency space introduced

in the previous example to define the derivatives of a function in L2(0, 2π) in aweak or generalized sense Let u ∈ C1(R), 2π−periodic The Fourier coefficients

(u)2= 2π

m∈Z

m2|?um|2 (6.15)

Thus, both sequences {?um} and {m?um} belong to l2

C But the right hand side in(6.15) does not involve u directly, so that it makes perfect sense to define

H1

per(0, 2π) =%

u∈ L2(0, 2π) :{?um} , {m?um} ∈ l2&

and introduce the inner product

per(0, 2π) is associated the function v∈ L2(0, 2π) given by

?umeimx= 1

mm|?um| ≤ 1

2

1

(uv+ uv)

316 6 Elements of Functional Analysis

6.4 Projections and Bases

6.4.1 Projections

Hilbert spaces are the ideal setting to solve problems in infinitely many dimensions.They unify through the inner product and the induced norm, both an analyticaland a geometric structure As we shall shortly see, we may coherently introducethe concepts of orthogonality, projection and basis, prove a infinite-dimensionalPythagoras’ Theorem (an example is just Bessel’s equation) and introduce otheroperations, extremely useful from both a theoretical and practical point of view

As in finite-dimensional linear spaces, two elements x, y belonging to an innerproduct space are called orthogonal or normal if (x, y) = 0, and we write x⊥y.Now, if we consider a subspace V ofRn, e.g a hyperplane through the origin,every x∈ Rn has a unique orthogonal projection on V In fact, if dimV = k andthe unit vectors v1, v2, , vkconstitute an orthonormal basis in V , we may alwaysfind an orthonormal basis inRn, given by

v1, v2, , vk, wk+1, , wn,where wk+1, , wn are suitable unit vectors Thus, if

In this case, the “infimum” in (6.16) is actually a “minimum”

The uniqueness of PVx follows from the fact that, if y∗∈ V and

|y∗−x| = |PVx− x| ,

6.4 Projections and Bases 317

then we must have

k

j=1

(y∗j− xj)2= 0,whence yj∗= xj for j = 1, , k, and therefore y∗= PVx Since

(x− PVx)⊥v, ∀v ∈ Vevery x∈ Rn may be written in a unique way in the form

x = y + zwith y∈ V and z ∈ V⊥, where V⊥denotes the subspace of the vectors orthogonal

to V

Then, we say that Rn is direct sum of the subspaces V and V⊥ and we write

Rn= V ⊕ V⊥.Finally,

|x|2

=|y|2

+|z|2

which is the Pythagoras’ Theorem inRn

Fig 6.1 Projection Theorem

We may extend all the above consideration to infinite-dimensional Hilbertspaces H, if we consider closed subspaces V of H Here closed means withrespect to the convergence induced by the norm More precisely, a subset U ⊂ H

is closed in H if it contains all the limit points of sequences in U Observe that if

V has finite dimension k, it is automatically closed, since it is isomorphic toRk

(orCk) Also, a closed subspace of a Hilbert space is a Hilbert space as well, withrespect to the inner product in H

Unless stated explicitly, from now on we consider Hilbert spaces over R(real Hilbert spaces), endowed with inner product (·, ·) and induced norm ·

318 6 Elements of Functional Analysis

Theorem 6.2 (Projection Theorem) Let V be a closed subspace of a Hilbertspace H Then, for every x∈ H, there exists a unique element PVx∈ V such that

PVx− x = inf

v∈Vv − x (6.17)Moreover, the following properties hold:

d = inf

v∈Vv − x

By the definition of least upper bound, we may select a sequence{vm} ⊂ V , suchthatvm− x → d as m → ∞ In fact, for every integer m ≥ 1 there exists vm∈ Vsuch that

vk− vm → 0

as well This proves that{vm} is a Cauchy sequence

Since H is complete, vm converges to an element w∈ H which belongs to V ,because V is closed Using the norm continuity (Proposition 6.1) we deduce

vm− x → w − x = d

so that w realizes the minimum distance from x among the elements in V

6.4 Projections and Bases 319

We have to prove the uniqueness of w Suppose ¯w∈ V is another element suchthat ¯w− x = d The parallelogram law, applied to the vectors w − x and ¯w− x,yields

w − ¯w2= 2w − x2+ 2 ¯w− x2− 4@@

@@w + ¯2 w− x@@

@@2

≤ 2d2+ 2d2− 4d2= 0whence w = ¯w

We have proved that there exists a unique element w = PVx∈ V such that

The elements PVx, QVx are called orthogonal projections of x on V and

V⊥, respectively The least upper bound in (6.17) is actually a minimum Moreoverthanks to properties 1, 2, we say that H is direct sum of V and V⊥ :

H = V ⊕ V⊥.Note that

V⊥={0} if and only if V = H

320 6 Elements of Functional Analysis

Remark 6.3 Another characterization of PVx is the following (see Problem 6.4):

(y, x) = lim (yn, x) = 0whence y∈ V⊥

Example 6.5 Let Ω ⊂ Rn be a set of finite measure Consider in L2(Ω) the1−dimensional subspace V of constant functions (a basis is given by f ≡ 1, forinstance) Since it is finite-dimensional, V is closed in L2(Ω) Given f ∈ L2(Ω),

to find the projection PVf, we solve the minimization problem

min

λ∈R Ω(f− λ)2.Since

Thus, the subspace V⊥ is given by the functions g∈ L2(Ω) with zero mean value

In fact these functions are orthogonal to f ≡ 1:

(g, 1)0=

Ω

g = 0

6.4.2 Bases

A Hilbert space is said to be separable when there exists a countable dense subset

of H An orthonormal basis in a separable Hilbert space H is sequence{wk}k≥1⊂

6.4 Projections and Bases 321

and every x∈ H may be expanded in the form

constitutes an orthonormal basis in L2(0, 2π) (see Appendix A)

It turns out that:

Proposition 6.2 Every separable Hilbert space H admits an orthonormal basis.Proof (sketch) Let {zk}k≥1 be dense in H Disregarding, if necessary, thoseelements which are spanned by other elements in the sequence, we may assumethat {zk}k≥1 constitutes an independent set, i.e every finite subset of{zk}k≥1 iscomposed by independent elements

Then, an orthonormal basis {wk}k≥1 is obtained by applying to{zk}k≥1 thefollowing so called Gram-Schmidt process First, construct by induction a sequence{ ˜w}k≥1 as follows Let ˜w1 = z1 Once ˜wk−1 is known, we construct ˜wk by sub-tracting from zk its components with respect to ˜w1, , ˜wk−1:

In the applications, orthonormal bases arise from solving particular boundaryvalue problems, often in relation to the separation of variables method Typicalexamples come form the vibrations of a non homogeneous string or from diffusion

322 6 Elements of Functional Analysis

in a rod with non constant thermal properties cv, ρ, κ The first example leads tothe wave equation

in [a, b] and positive in (a, b)

In general, the resulting boundary value problem has non trivial solutions onlyfor particular values of λ, called eigenvalues The corresponding solutions are calledeigenfunctions and it turns out that, when suitably normalized, they constitute anorthonormal basis in the Hilbert space L2

w(a, b), the set of Lebesgue measurablefunctions in (a, b) such that

u2

L 2w =

b a

u2(x) w (x) dx <∞,endowed with the inner product

(u, v)L2w=

b a

u (x) v (x) w (x) dx

We list below some examples5

• Consider the problem

6.4 Projections and Bases 323

which shows the proper weight function w (x) = 

1− x2−1/2

The eigenvaluesare λn = n2, n = 0, 1, 2, The corresponding eigenfunctions are the Chebyshevpolynomials Tn, recursively defined by T0(x) = 1, T1(x) = x and

Tn+1= 2xTn− Tn−1 (n > 1) For instance:



1− x2

u(x)→ 0 as x→ ±1

The differential equation is known as Legendre’s equation The eigenvalues are

λn = n (n + 1), n = 0, 1, 2, The corresponding eigenfunctions are the Legendrepolynomials, defined by L0(x) = 1, L1(x) = x,

2n + 1

2 Lnconstitute an orthonormal basis in L2(−1, 1) (here w (x) ≡ 1) Every function

324 6 Elements of Functional Analysis

The differential equation is known as Hermite’s equation (see Problem 6.6) andmay be written in the form (6.21):

(e−x2u)+ 2λe−x2u = 0which shows the proper weight function w (x) = e−x2 The eigenvalues are λn =

n, n = 0, 1, 2, The corresponding eigenfunctions are the Hermite polynomialsdefined by Rodrigues’ formula

Hn(x) = (−1)nex2 d

n

dxne−x2 (n≥ 0) For instance

H0(x) = 1, H1(x) = 2x, H2(x) = 4x2− 2, H3(x) = 8x3− 12x.The normalized polynomials π−1/4(2nn!)−1/2Hn constitute an orthonormal basis

u (0) finite, u (a) = 0 (6.23)Equation (6.22) may be written in Sturm-Liouville form as

(xu)+



λx−p2x



u = 0which shows the proper weight function w (x) = x The simple rescaling z =√

λxreduces (6.22) to the Bessel equation of order p

where the dependence on the parameter λ is removed The only bounded solutions

of (6.24) are the Bessel functions of first kind and order p, given by

p+2k

6.4 Projections and Bases 325

Fig 6.2 Graphs of J0,J1 and J2

where

Γ (s) =

∞ 0

a x

 The normalized eigenfunctions

√2

aJp+1(αpj)Jp

αpj

a x



constitute an orthonormal basis in L2

w(0, a), with w (x) = x Every function f ∈

where

fj= 2

a2J2 p+1(αpj)

a 0

xf (x) Jpαpj

a x

dx,convergent in L2w(0, a)

326 6 Elements of Functional Analysis

6.5 Linear Operators and Duality

R (L) = {y ∈ H2:∃x ∈ H1, Lx = y}

N (L) and R (L) are linear subspaces of H1 and H2, respectively

Our main objects will be linear bounded operators

Definition 6.4 A linear operator L : H1 → H2 is bounded if there exists anumber C such that

LxH2 ≤ C xH1, ∀x ∈ H1 (6.26)The number C controls the expansion rate operated by L on the elements of

H1 In particular, if C < 1, L contracts the sizes of the vectors in H1

If x= 0, using the linearity of L, we may write (6.26) in the form

Proposition 6.3 A linear operator L : H1→ H2 is bounded if and only if it iscontinuous

7 Notation: if L is linear, when no confusion arises, we may write Lx instead of L (x)

6.5 Linear Operators and Duality 327

Proof Let L be bounded From (6.26) we have,∀x, x0∈ H1,

LyH2 ≤ 1

δand (6.27) holds with K≤ C = 1

δ.Given two Hilbert spaces H1 and H2, we denote by

L (H1, H2)the family of all linear bounded operators from H1 into H2 If H1= H2we simplywriteL (H) L (H1, H2) becomes a linear space if we define, for x∈ H1and λ∈ R,

(G + L) (x) = Gx + Lx(λL) x = λLx

Also, we may use the number K in (6.27) as a norm inL (H1, H2):

Proposition 6.4 Endowed with the norm (6.28), L (H1, H2) is a Banach space.Example 6.6 Let A be an m× n real matrix The map

L : x−→ Ax

is a linear operator fromRn into Rm To computeL, note that

Ax2= Ax· Ax = A Ax· x

328 6 Elements of Functional Analysis

The matrix A A is symmetric and nonnegative and therefore, from Linear bra,

Alge-sup

x=1A Ax· x =ΛM

where ΛM is the maximum eigenvalue of A A Thus,L =√ΛM

Example 6.7 Let V be a closed subspace of a Hilbert space H The projections

x−→ PVx, x−→ QVx,defined in Theorem 6.2, are bounded linear operators from H into H In fact, from

x2=PVx2+QVx2, it follows immediately that

PVx ≤ x , QVx ≤ x

so that (6.26) holds with C = 1 Since PVx = x when x∈ V and QVx = x when

x∈ V⊥, it follows thatPV = QV = 1 Finally, observe that

N (PV) =R (QV) = V⊥ and N (QV) =R (PV) = V

Example 6.8 Let V and H be Hilbert spaces with8V ⊂ H Considering an element

in V as an element of H, we define the operator IV→H: V → H,

IV →H(u) = u,which is called embedding of V into H IV→H is clearly a linear operator and it

is also bounded if there exists a constant C such that

uH≤ C uV , for every u∈ V

In this case, we say that V is continuously embedded in H and we write

V → H

For instance, H1

per(0, 2π) → L2(0, 2π)

6.5.2 Functionals and dual space

When H2=R (or C, for complex Hilbert spaces), a linear operator L : H → Rtakes the name of functional

Definition 6.5 The collection of all bounded linear functionals on a Hilbert space

H is called dual space of H and denoted by H∗ (instead ofL (H, R))

8 The inner products in V and H may be different

6.5 Linear Operators and Duality 329

Example 6.9 Let H = L2(Ω), Ω⊆ Rnand fix g∈ L2(Ω) The functional definedby

Example 6.10 The functional in Example 6.13 is induced by the inner productwith a fixed element in L2(Ω) More generally, let H be a Hilbert space For fixed

y∈ H, the functional

L1: x−→ (x, y)

is continuous In fact Schwarz’s inequality yields|(x, y)| ≤ x y, whence L1∈

H∗andL1 ≤ y Actually L1 = y since, choosing x = y, we have

h∈ Rn,

Lh = a· handL = |a| The following theorem says that an analogous result holds in Hilbertspaces

Theorem 6.3 (Rieszs Representation Theorem) Let H be a Hilbert space Forevery L∈ H∗ there exists a unique uL∈ H such that:

1 Lx = (uL, x) for every x∈ H,

2.L = uL

330 6 Elements of Functional Analysis

Proof Let N be the kernel of L If N = H, then L is the null operator and

uL = 0 IfN ⊂ H, then N is a closed subspace of H In fact, if {xn} ⊂ N and

xn→ x, then 0 = Lxn → Lx so that x ∈ N ; thus N contains all its limit pointsand therefore is closed

Then, by the Projection Theorem, there exists z ∈ N⊥, z= 0 Thus Lz = 0and, given any x∈ H, the element

w = x−Lx

Lzzbelongs toN In fact

For the uniqueness, observe that, if v∈ H and

Lx = (v, x) for every x∈ H,subtracting this equation from Lx = (uL, x), we infer

(uL− v, x) = 0 for every x∈ Hwhich forces v = uL

To showL = uL, use Schwarz’s inequality

uL ≤ L ThusL = uL 

6.5 Linear Operators and Duality 331

The Riesz’s map R : H∗→ H given by

Typically, L2(Ω) or l2 are identified with their duals

Remark 6.5 Warning: there are situations in which the above canonical cation requires some care A typical case we shall meet later occurs when dealingwith a pair of Hilbert spaces V , H such that

identifi-V → H and H∗→ V∗

As we will see in subsection 6.8.1, in this conditions it is possible to identify Hand H∗and write

V → H → V∗,but at this point the identification of V with V∗ is forbidden, since it would giverise to nonsense!

Remark 6.6 A few words about notations The symbol (·, ·) or (·, ·)H denotesthe inner product in a Hilbert space H Let now L ∈ H∗ For the action of thefunctional L on an element x∈ H we used the symbol Lx Sometimes, when it isuseful or necessary to emphasize the duality (or pairing) between H and H∗, weshall use the notationL, x∗ or evenH∗L, xH

6.5.3 The adjoint of a bounded operator

The concept of adjoint operator extends the notion of transpose of an m× nmatrix A and plays a crucial role in determining compatibility conditions for thesolvability of several problems The transpose A is characterized by the identity

(Ax, y)Rm=(x, A y)Rn, ∀x ∈Rn,∀y ∈Rm

We extend precisely this relation to define the adjoint of a bounded linear operator.Let L∈ L(H1, H2) If y∈ H2 is fixed, the real map

Ty : x−→ (Lx,y)H

332 6 Elements of Functional Analysis

defines an element of H1∗ In fact

|Tyx| =(Lx,y)H

2 ≤ LxH2yH2 ≤ LL(H1,H2)yH2xH1

so thatTy ≤ LL(H1,H2)yH2

From Riesz’s Theorem, there exists a unique w ∈ H1 depending on y, which

we denote by w = L∗y, such that

Tyx = (x,L∗y)H1 ∀x ∈ H1,∀y ∈ H2.This defines L∗as an operator from H2 into H1, which is called the adjoint of L.Precisely:

Definition 6.6 The operator L∗: H2→ H1 defined by the identity

(Lx,y)H

2= (x,L∗y)H

1, ∀x ∈ H1,∀y ∈ H2 (6.30)

is called the adjoint of L

Example 6.11 Let R : H∗ → H the Riesz operator Then R∗ = R−1 : H → H∗

In fact, for every F ∈ H∗ and v∈ H, we have:

(RF, v)H =F, v∗=

F, R−1v

H∗.Example 6.12 Let T : L2(0, 1)→ L2(0, 1) be the linear map

T u (x) =

x 0

u2)dx≤ 1

2

1 0

u2≤ 1

2u2 0

and therefore T is bounded To compute T∗, observe that

(T u,v)0=

1 0

[v (x)

x 0

u (y) dy] dx = exchanging the order of integration

=

1 0

[u (y)

1 x

v (x) dx] dy = (u, T∗v)0.Thus,

T∗v (x) =

1 x

v (t) dt

6.5 Linear Operators and Duality 333

Symmetric matrices correspond to selfadjoint operators We say that L is adjoint if H1= H2 and L∗= L Then, (6.30) reduces to

self-(Lx,y) = (x,Ly)

An example of a selfadjoint operator in a Hilbert space H is the projection PV on

a closed subspace of H; in fact, recalling the Projection Theorem:

(PVx,y) = (PVx,PVy + QVy) = (PVx,PVy) = (PVx + QVx,PVy) = (x,PVy) Important self-adjoint operators are associated with inverses of differential oper-ators, as we will see in Chapter 8

The following properties are immediate consequences of the definition of adjoint(for the proof, see Problem 6.10)

Proposition 6.5 Let L , L1∈ L(H1, H2) and L2∈ L(H2, H3) Then:

(a) L∗∈ L(H2, H1) Moreover L∗∗ = L and

L∗L(H2,H1)=LL(H1,H2).(b) (L2L1)∗= L∗1L∗2 In particular, if L is an isomorphism, then



L−1∗

= (L∗)−1.The next theorem extends relations well known in the finite-dimensional case.Theorem 6.4 Let L∈ L (H1, H2) Then

R (L)⊥⊆ N (L∗) ,equivalent to

N (L∗)⊥⊆ R (L)

b) letting L = L∗ in a) we deduce

R (L∗) =N (L)⊥,equivalent toR (L∗)⊥=N (L) 

9 Remark 6.8

334 6 Elements of Functional Analysis

6.6 Abstract Variational Problems

6.6.1 Bilinear forms and the Lax-Milgram Theorem

In the variational formulation of boundary value problems a key role is played bybilinear forms Given two linear spaces V1, V2, a bilinear form in V1× V2 is afunction

a : V1× V2→ Rsatisfying the following properties:

i) For every y∈ V2, the function

When V1= V2, we simply say that a is a bilinear form in V

Remark 6.7 In complex inner product spaces we define sesquilinear forms, instead

of bilinear forms, replacing ii) by:

iibis) for every x∈ V1, the function

y−→ a(x, y)

is anti-linear10 in V2

Here are some examples

• A typical example of bilinear form in a Hilbert space is its inner product

• The formula

a (u, v) =

b a

(p(x)uv+ q(x)uv + r(x)uv) dx

where p, q, r are bounded functions, defines a bilinear form in C1([a, b])

More generally, if Ω is a bounded domain inRn,

6.6 Abstract Variational Problems 335

Ω

• A bilinear form in C2

Ωinvolving higher order derivatives is

|a(u, v)| ≤ M u v , ∀u, v ∈ V ;ii) a is V−coercive, i.e there exists a constant α > 0 such that

a(v, v)≥ α v2, ∀v ∈ V, (6.32)then there exists a unique solution u ∈ V of problem (6.31) Moreover, thefollowing stability estimate holds:

u ≤ 1

αF V∗ (6.33)Remark 6.8 The coercivity inequality (6.32) may be considered as an abstractversion of the energy or integral estimates we met in the previous chapters Usually,

it is the key estimate to prove in order to apply Theorem 6.5 We shall come back

to the general solvability of a variational problem in Section 6.8, when a is not

V−coercive

Remark 6.9 Inequality (6.61) is called stability estimate for the following reason.The functional F, element of V∗, encodes the “data” of the problem (6.31) Sincefor every F there is a unique solution u(F ), the map

F−→ u(F )

336 6 Elements of Functional Analysis

is a well defined function from V∗ onto V Also, everything here has a linearnature, so that the solution map is linear as well To check it, let λ, μ ∈ R, F1,

F2∈ V∗and u1, u2 the corresponding solutions The bilinearity of a, gives

a(λu1+ μu2, v) = λa(u1, v) + μa(u2, v) =

= λF1v + μF2v

Therefore, the same linear combination of the solutions corresponds to a linearcombination of the data; this expresses the principle of superposition for problem(6.31) Applying now (6.33) to u1− u2, we obtain

u1− u2 ≤ 1

αF1− F2V∗.Thus, close data imply close solutions The stability constant 1/α plays an im-portant role, since it controls the norm-variation of the solutions in terms of thevariations on the data, measured byF1− F2V∗ This entails, in particular, thatthe more the coercivity constant α is large, the more “stable” is the solution.Proof of theorem 6.5 We split it into several steps

1 Reformulation of problem (6.31) For every fixed u ∈ V , by the continuity

of a, the linear map

v→ a (u, v)

is bounded in V and therefore it defines an element of V∗ From Riesz’s tation Theorem, there exists a unique A [u]∈ V such that

Represen-a (u, v) = (A[u],v) , ∀v ∈ V (6.34)Since F ∈ V∗ as well, there exists a unique zF ∈ V such that

F v = (zF,v) ∀v ∈ Vand moreoverF V∗ =zF Then, problem (6.31) can be recast in the following

⎨

⎩

Find u∈ Vsuch that(A [u] ,v) = (zF,v) , ∀v ∈ Vwhich, in turn, is equivalent to finding u such that

6.6 Abstract Variational Problems 337

2 Linearity and continuity of A We repeatedly use the definition of A and thebilinearity of a To show linearity, we write, for every u1, u2, v∈ V and λ1, λ2∈ R,(A [λ1u1+ λ2u2] ,v) = a (λ1u1+ λ2u2, v) = λ1a (u1, v) + λ2a (u2, v)

= λ1(A [u1] ,v) + λ2(A [u2] ,v) = (λ1A [u1] + λ2A [u2] ,v)whence

A [λ1u1+ λ2u2] = λ1A [u1] + λ2A [u2] Thus A is linear and we may write Au instead of A [u] For the continuity, observethat

Au2 = (Au, Au) = a(u, Au)

≤ M u Au

whence

Au ≤ M u

3 A is one-to-one and has closed range, i.e

N (A) = {0} and R (A) is a closed subspace of V

In fact, the coercivity of a yields

αu2≤ a (u, u) = (Au, u) ≤ Au u

whence

u ≤ 1

Thus, Au = 0 implies u = 0 and henceN (A) = {0}

To prove that R (A) is closed we have to consider a sequence {ym} ⊂ R (A)such that

um→ uand the continuity of A yields ym= Aum→ Au Thus Au = y, so that y ∈ R (A)andR (A) is closed

338 6 Elements of Functional Analysis

4 A is surjective, that is R (A) = V Suppose R (A) ⊂ V Since R (A) is aclosed subspace, by the Projection Theorem there exists z = 0, z ∈ R (A)⊥ Inparticular, this implies

0 = (Az, z) = a (z, z)≥ α z2

whence z = 0 Contradiction ThereforeR (A) = V

5 Solution of problem (6.31) Since A is one-to-one and R (A) = V , thereexists exactly one solution u∈ V of equation

Au = zF.From point 1, u is the unique solution of problem (6.31) as well

6 Stability estimate From (6.36) with u = u, we obtain

u ≤ 1

αAu = 1

αzF = 1

αF V∗

and the proof is complete.

Remark 6.10 Some applications require the solution to be in some Hilbert space

W , while asking the variational equation

a (u, v) =F, v∗

to hold for every v ∈ V , with V = W A variant of Theorem 6.5 deals with thisasymmetric situation Let F ∈ V∗ and a = a (u, v) be a bilinear form in W× Vsatisfying the following three hypotheses:

i) there exists M such that

|a(u, v)| ≤ M uWvV , ∀u ∈ W, ∀v ∈ V ;ii) there exists α > 0 such that

Condition ii) is an asymmetric coercivity, while iii) assures that, for every fixed

v∈ V , a (v, ·) is positive at some point in W We have (for the proof see Problem6.11):

Theorem 6.6 (Ne¸cas) If i), ii), iii) hold, there exists a unique u∈ W such that

a (u, v) =F, v∗ ∀v ∈ V

Moreover

uW ≤ 1

αF V∗ (6.37)

6.6 Abstract Variational Problems 339

6.6.2 Minimization of quadratic functionals

When a is symmetric, i.e if

a (u, v) = a (v, u) ∀u, v ∈ V,the abstract variational problem (6.31) is equivalent to a minimization problem

In fact, consider the quadratic functional

Letting ϕ (ε) = E (u + εv), from the above calculations we have

ϕ(0) = a (u, v)− F, v∗

340 6 Elements of Functional Analysis

Thus, the linear functional

v−→ a (u, v) − F, v∗appears as the derivative of E at u along the direction v and we write

E(u) v = a (u, v)− F, v∗ (6.39)

In Calculus of Variation E is called first variation and denoted by δE

If a is symmetric, the variational equation

E(u) v = a (u, v)− F, v∗= 0, ∀v ∈ V (6.40)

is called Euler equation for the functional E

Remark 6.11 A bilinear form a, symmetric and coercive, induces in V the innerproduct

(u, v)a = a (u, v)

In this case, existence, uniqueness and stability for problem (6.31) follow directlyfrom Riesz’s Representation Theorem In particular, there exists a unique mini-mizer u of E

6.6.3 Approximation and Galerkin method

The solution u of the abstract variational problem (6.31), satisfies the equation

a (u, v) =F, v∗ (6.41)for every v in the Hilbert space V In concrete applications, it is important tocompute approximate solutions with a given degree of accuracy and the infinitedimension of V is the main obstacle Often, however, V may be written as a union

of finite-dimensional subspaces, so that, in principle, it could be reasonable toobtain approximate solutions by “projecting” equation (6.41) on those subspaces.This is the idea of Galerkin’s method In principle, the higher the dimension

of the subspace the better should be the degree of approximation More precisely,the idea is to construct a sequence {Vk} of subspaces of V with the followingproperties:

a) Every Vk is finite-dimensional : dimVk= k,

b) Vk ⊂ Vk+1(actually, not strictly necessary),

c) ∪Vk = V

To realize the projection, assume that the vectors ψ1, ψ2, , ψk span Vk Then,

we look for an approximation of the solution u in the form

6.6 Abstract Variational Problems 341

by solving the projected problem

a (uk, v) =F, v∗ ∀v ∈ Vk (6.43)Since{ψ1, ψ2, , ψk} constitutes a basis in Vk, (6.43) amounts to requiring

a (uk, ψr) =F, ψr∗ r = 1, , k (6.44)Substituting (6.42) into (6.44), we obtain the k linear algebraic equations

we may write (6.45) in the compact form

The matrix A is called stiffness matrix and clearly plays a key role in the numericalanalysis of the problem

If the bilinear form a is coercive, A is strictly positive In fact, letξ ∈Rk Then,

by linearity and coercivity:

A is strictly positive and, in particular, non singular

x2< /small>=PVx2< /small>+QVx2< /small>, it... (6.44)Substituting (6. 42) into (6.44), we obtain the k linear algebraic equations

we may write (6.45) in the compact form

The matrix A is called stiffness matrix and clearly plays a key role in. .. μF2< /sub>v

Therefore, the same linear combination of the solutions corresponds to a linearcombination of the data; this expresses the principle of superposition for problem(6.31) Applying