c hapter 1 distr ibutions on op en sets of r d - upmc · c hapter 1 distr ibutions on op en sets of...
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Chapter 1
Distributions on open sets of Rd
”. . . For the Fourier transform, the introduction of distribution (hencethe space S) is inevitable either in an explicit or hidden form. . .As aresult one may obtain all that is desired from the point of view of thecontinuity and inversion of the Fourier transform.”Laurent Schwartz (1915-2002)
The theory of distributions has been introduced by L. Schwartz (1915-2002) ca. 1944-1950 as ”ob-jects” which generalize functions [Schwartz, 1945, Schwartz, 1966]. They extend the notion of derivativeto all integrable functions and are now widely used to formulate generalized solutions of partial di!er-ential equations. The Lebesgue Lp spaces contain non regular and non continuous functions for whichderivatives are not defined in the classical sense. Nonetheless, the classical derivatives generally existalmost everywhere. It seems then reasonable to generalize the notion of derivative to be independent ofzero-measure subsets. This leads to the concept of weak derivative introduced by J. Leray (1906-1998)and S.L. Sobolev (1908-1989). The main concept underlying the notion of weak derivative is the conceptof distributions.
Contents1.1 Infinitely smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The concept of distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
According to [Schwartz, 1966], O. Heaviside (1850-1925) introduced in 1894 a function denoted H andcalled the unit step function. It corresponds to the characteristic function of the interval [0,+![ of R andis a discontinuous function whose value is zero for negative argument and one for positive argument (cf.Figure 1.1). This function has no derivative at the origin. Nonetheless, Heaviside defined the derivativeH ! of H at any point in R, and called it the unit impulse function, as:
H !(x) =!
0 x "= 0+! x = 0 (1.1)
This function H cannot be considered as a usual function. Since H ! vanishes almost everywhere, it shouldbe assigned integral zero: " a
"aH !(x) dx = 0 , #a $]0,+
4 Chapter 1. Distributions on open sets of Rd
Furthermore, we have:" a
"aH !(x) dx = H(a)%H(%a) = 1 , #a $]0,+
1.1. Infinitely smooth functions 5
Definition 1.1 The function f : " & Rd ( Rm is said to be di!erentiable at x $ " if there exists anelement f !(x) $ L(Rd, Rm) such that
'f(x + h)% f(x)% f !(x)h' = o('h') , h ( 0 .
Lemma 1.1 If f : " ( R is di!erentiable at x0, then f has a partial derivative with respect to xj at x0
and"f
"xj(x0) = f !(x0)ej , #j $ {1, . . . , d} ,
where (e1, . . . , ed) denotes the canonical basis of Rd. Moreover, we have
f !(x0)v = )*f(x0), v+ , #v $ Rd ,
the vector *f(x0) =d#
j=1
"f
"xj(x0)ej is the gradient of f at x0,
Proposition 1.1 Suppose I is an open interval of R and f is continuous in I and di!erentiable exceptat point x0 $ I. If x $ I and lim
x%x0f !(x) = a $ R, then f !(x0) exists and we have f !(x0) = a.
Definition 1.2 Consider an open subset " & Rd and k $ N. A function f $ Ck(") if one of thefollowing conditions holds:
(i) k = 0 and f is continuous on ",
(ii) k , 1 and f has partial derivatives"f
"xj, j = 1, . . . , d that are in Ck"1(").
Moreover, f $ C#(") if f $ Ck(") for all k $ N.
Lemma 1.2 (Schwarz) If f $ C2(") then
"2f
"xi"xj=
"2f
"xj"xi, #i, j $ {1, . . . , d} .
Notice that if f $ Ck(", this lemma refers to the property of interchanging the order of taking partialderivatives of a function. The matrix of second-order partial derivatives of f is called the Hessian matrix.In most cases, this matrix is a symmetric matrix.
Theorem 1.1 (Taylor’s formula) Let " be an open subset of Rd, m $ N and f $ Cm+1("). Considerx, y $ " such that x + ty $ ", for all t $ [0, 1]. Then, we have:
f(x + y) =#
|!|&m
y!
#!"!f(x) + (m + 1)
#
|"|=m+1
y"
$!
" 1
0(1% t)m""f(x + ty) dt . (1.4)
Corollary 1.1 Let " be an open subset of Rd, f $ Cm+1(") and K a compact subset of ". Considerx, y $ K such that [x, x + y] & K. Then, there exists a constant C = C(K, m, f) such that
$$$$$$f(x + y)%
#
|!|&m
y!
#!"!f(x)
$$$$$$- C'y'm+1 . (1.5)
Corollary 1.2 If f $ Ck(B) where B = {x $ Rd, 'x' < 1}, then there exists f1, . . . , fd $ Ck"1(B) suchthat:
"!fj(0) ="!"jf(0)1 + |#| , sup
x'B|"!fj | - sup
x'B|"!"jf | , #j = 1, . . . , d f(x)% f(0) =
d#
j=1
xjfj(x) .
6 Chapter 1. Distributions on open sets of Rd
1.1.2 Existence of C# functions
Let " & Rd be an open set. We recall that the set of infinitely smooth functions is defined as:
C#(") =%
k(0
Ck(") .
Definition 1.3 Given u $ C0("), the support of u, denoted as Supp(u), is defined as the closure of theset {x $ " , u(x) "= 0} in ". It is the smallest closed subset of " such that u = 0 in "\Supp(u).
Notice that if x $ Supp(u), then there exists a sequence (xn)n'N & " such that u(xn) "= 0 for all n $ Nand such that limn%# xn = x.
Definition 1.4 If k $ N . {+!}, the space Ck0 (") is composed of all functions u $ Ck(") having a
compact subset of " as support. The elements of C#0 ("), hereafter denoted by D("), are called testfunctions.
It is common to find the notations C#0 (") or C#c (") instead of the symbol D("). Every functionu $ Ck
0 (") can be extended to a function of Ck0 (Rd). Thus, Ck
0 (") can be seen as a subspace of Ck0 (Rd).
In this respect, given an open set " & Rd, the set Ck0 (") can be defined as the set of elements u $ Ck
0 (Rd)for which Supp(u) & ".
The space D(") is not empty. Indeed, we have the following result.
Lemma 1.3 There exists a function u $ D(Rd) such that u(0) > 0 and u(x) , 0, for all x $ Rd.
Proof. Consider the function f $ C!(R) defined as:
f(x) =!
0 x - 0exp (%1/x) x > 0
Then, the functionu(x) = f(1% 'x'2)
satisfies the assumptions. By translation and scaling, we show that for every r > 0, the function
x /( u
&x% x0
r
'
is positive on R, strictly positive at x0 and its support is the ball of radius r centered at x0. !The existence of such function u allows to prove a classical result.
Theorem 1.2 If f, g $ C0(") and if the following identity holds"
!fu dx =
"
!gu dx , #u $ D(") ,
then, f = g.
Proof. Let consider h = f % g, then"
!hu dx = 0 , #u $ D(") .
If h is a complex valued function, we will consider the real and imaginary parts separately. Hence, consider h is areal valued function and that the previous equality holds for all u $ D("), u being a real valued function. If thereexists x0 such that h(x0) "= 0 then we select u such that u(x0) > 0 with support in a neighborhood of x0 and suchthat uh has constant sign. This obviously is in contradiction with the assumption, thus h 0 0 in ". !Lemma 1.4 There exists an increasing function % $ C#(R) such that
%(x) =!
0 x - 01 x , 1
1.1. Infinitely smooth functions 7
Proof. Consider the function f $ C!(R) defined hereabove as:
f(x) =!
0 x - 0exp (%1/x) x > 0
In this case, Supp(f) = [0,![ and 0 - f(x) - 1, for all x $ R. We introduce the functions g(x) = f(x)f(1%x) andG(x) =
( x0 g(t) dt. We notice that 0 - g(x) - 1, for all x $ R and that Supp(g) & [0, 1] and g "0 0. The function
% =G(x)G(1)
$ C!(R) is an increasing function and is such that
%(x) =!
0 x - 01 x , 1
!
Proposition 1.2 Consider four real numbers a, b, c and d such that a < c < d < b. Then, there exists& $ D(R) such that:
(i) &(x) = 1, for all x $]c, d[,
(ii) Supp(&) &]a, b[,
(iii) 0 - &(x) - 1, for all x $ R.
Proof. Consider the function &(x) = %
&x% a
c% a
'%
&b% x
b% d
', where % is the function defined in the previous
lemma. !
If K is a compact subset of Rd and ' > 0, we denote
K# = K + B(0, ') =)
x'K
B(x, ') ,
where B(x, ') the ball of radius ' centered at x $ Rd.
Proposition 1.3 Consider a compact set K & Rd. Then, for every ' > 0, there exists u $ D(K2#) suchthat: !
u(x) = 1 #x $ K#
0 - u(x) - 1 #x $ Rd
Proof. By compacity, there exists x1, . . . , xp $ K such that
K! &p)
j=1
B
&xj ,
4'
3
'.
For each j = 1, . . . , p there exists a function uj $ D(B(xj ,5!3 )) such that uj(x) , 0 for all x $ Rd and uj 0 1 on
B(xj ,4!3 ). Consider u(x) =
*dj=1 uj(x). Then, u(x) , 1 for all x $
p)
j=1
B(xj , '). Moreover, given that
p)
j=1
B
&xj ,
5'
3
'& K2! ,
we conclude that u $ D(K2!). Taking u = %(u(x)) where % $ C!(R) is the function introduced in the previouslemma, yields the result. !
8 Chapter 1. Distributions on open sets of Rd
Corollary 1.3 Let K1, K2 be two disjoined compact subsets of the open set " & Rd. Then, there existsa function u $ D(") such that
u(x) =!
1 #x $ K1
%1 #x $ K2
and such that |u(x)| - 1 for all x $ ".
Proof. Consider U1 and U2 two open sets of " such that
K1 & U1 , K2 & U2 , U1 1 U2 = 2 .
From the previous proposition, we know that there exists u1, u2 $ D(") such that:
ui 0 1 , in Ki , ui $ D(Ui) , 0 - ui(x) - 1 , i $ {1, 2} .
The function u defined as: u(x) = u1(x)% u2(x), #x $ " satisfies the desired properties. !
1.1.3 Partition of unity
Partitions of unity are useful because they often allow one to extend local constructions to the wholespace.
Proposition 1.4 Let K be a compact subset of Rd and let consider an open cover (Uj)j=1,...,N of K.Then, there exists compacts sets (Kj)j=1,...,N such that Kj & Uj, for all j = 1, . . . , N and
K =N)
j=1
Kj . (1.6)
Proof. For every x $ K, consider rx > 0 such that B(x, rx) &%
x"Uj
Uj . Hence, we have K &)
x"K
B(x, rx) and
thus there exists x1, . . . , xM $ K such that K &M)
i=1
B(xi, rxi). We pose
Kj = K 1
+
,)
B(xi,rxi )#Uj
B(xi, rxi)
-
. .
Then by definition, Kj is a compact set included in K such that Kj & Uj . Let consider x $ K. There existsi $ {1, . . . ,M} such that x $ B(xi, rxi). Moreover, there exists j0 $ {1, . . . , N} such that xi $ Uj0 , thusB(xi, rxi) & Uj0 . We conclude that x $ Kj0 . !
Theorem 1.3 (Partition of unity) Consider a compact subset K of Rd and open sets Uj of Rd. Sup-
pose we have K &N)
j=1
Uj, then there exists a collection (uj)j=1,...,N such that uj $ D(Uj), 0 - uj - 1 for
all j = 1, . . . , N and such thatN#
j=1
uj = 1 in the neighborhood of K.
1.1. Infinitely smooth functions 9
Proof. We know that there exists compact sets (Kj)j"{1,...,N} such that Kj & Uj , for every j = 1, . . . , N and
such that K =N)
j=1
Kj . Moreover, from the previous proposition, we deduce that for every j = 1, . . . , N , there
exists vj $ D(Uj) such that vj(x) $ [0, 1], for all x $ Rd and vj(x) = 1 if x $ Kj . Consider the open set
V =
/0
1x $N)
j=1
Uj ,N#
j=1
vj(x) > 0
23
4 .
We have then K & V . Hence, there exists ( $ D(V ) such that ((x) $ [0, 1] for x $ Rd and ( 0 1 on an open setW such that K & W & V . Considering the function
ui =vj
(1% () +*N
k=1 vk
,
we notice that ui $ D(Uj) as the denominator is strictly positive on V and is equal to 1 outside V . Since ( 0 1 on
the set W , the relation implies thatN#
j=1
ui 0 1 on W . !
Definition 1.5 Under the hypothesis of Theorem 1.3, the collection (uj)j=1,...,N is called a partition ofunity subordinate to the open cover (Uj)j=1,...,N of K.
1.1.4 Lebesgue integration
We briefly recall some important results about Lebesgue integration and Lp spaces and we refer thereader to Appendix D for more details and results.
If " is an open subset of Rd and p , 1, we denote by Lp(") the space of p-power integrable functionswith values in R. The space Lp(") endowed with the norm
'f'Lp(!) =&"
!|f(x)|p dx
' 1p
, #f $ Lp(") , (1.7)
is a Banach space. If p , 1, we denote by Lploc(") the space of locally integrable functions f such that
f $ Lp(K), for every compact K $ ".
Lemma 1.5 If f $ Lploc(") then, there exists a largest open set V in " such that f |V = 0 almost
everywhere in V .
Definition 1.6 Consider " & Rd and f $ L1loc("). We call essential support of f the closed set "\V
where V is the largest open V & " such that f |V = 0 almost everywhere in V .
Theorem 1.4 (Lebesgue dominated convergence) Let (fn)n'N be a sequence of functions of L1("),with " an open set of Rd. Suppose that
(i) fn(x) ( f(x) almost everywhere on ",
(ii) there exists a function g $ L1(") such that for every n, |fn(x)| - g(x) almost everywhere in ".
Then, f $ L1(") and limn%# 'fn % f'L1(!) = 0.
Theorem 1.5 (Lebesgue) Let " be an open set of Rd, f : Rp 3 " ( R and k $ N. Suppose that
(i) the function fx : " ( R defined by fx()) = f(x,)) belongs to Ck(") for every fixed x $ Rp,
10 Chapter 1. Distributions on open sets of Rd
(ii) for all # $ Nn such that |#| - k we have
sup$'!
|"!fx())| - w!(x) ,
where w! $ L1(Rp).
Then, the function g()) ="
Rpf(x,)) dx is a function of Ck(") and for every ) $ R we have:
"!g()) ="
Rp"!
$ f(x,)) dx , ## $ Nn |#| - k .
Consider "1 & Rn1 , "2 & Rn2 two open sets and a measurable function F : "1 3 "2 ( C. We havetwo classical results.
Theorem 1.6 (Tonelli) Suppose that"
!2
|F (x, y)| dy < ! for almost every x $ "1 and that
"
!1
dx
"
!2
|F (x, y)| dy < ! .
Then F $ L1("1 3 "2).
Theorem 1.7 (Fubini) Suppose F $ L1("1 3 "2). Then, we have F (x, y) $ L1("2) for almost every
x $ "1 and"
!2
F (x, y) dy $ L1("1). Similarly, we have F (x, y) $ L1("1) for almost every y $ "2 and"
!1
F (x, y) dx $ L1("2). Furthermore, we have:
"
!1
dx
"
!2
F (x, y) dy ="
!2
dy
"
!1
F (x, y) dx ="
!1)!2
F (x, y) dxdy .
To conclude this section, we give the following density result.
Theorem 1.8 (i) If p , 1, then the space C00 (") is dense in Lp(").
(ii) If 1 - p < !, the space D(") is dense in Lp(").
1.2 The concept of distribution
1.2.1 Main definitions
Let " be an open subset of Rd and let K denote a compact of ". In this section, we call test function on "any function of D(") and we denote by DK(") the set of test functions with support in K. Furthermore,we like to introduce the following notations: given (#,*) $ Nd 3 Nd, x $ Rd, and u $ C#(") :
|#| def=d#
j=1
#j , #! def=d5
j=1
#j ! , x! def=d5
j=1
x!j
j , "!udef=
d5
j=1
"!j
j u .
1.2. The concept of distribution 11
Definition 1.7 We call distribution on " any linear form T on D(") such that the following continuityproperty holds: for every compact set K & ", there exists an integer N and a constant CK such that:
|)T, u+| - CK sup|!|&N
'"!u'L! , #u $ DK(") (1.8)
and we denote by D!(") the set of all distributions on ".
If for every compact K & ", the previous inequality holds for the same integer N , then the order of thedistribution u is at most N .
Example 1.1 (i) Every function f $ L1loc(") defines a distribution Tf $ D!(") or order 0 by posing
)Tf , u+ ="
!f(x)u(x) dx , #u $ D(") .
(ii) Given a $ ", we consider the linear form on D(") defined as
)!a, u+ = u(a) , #u $ D(") .
Hence, we have|)!a, u+| - 'u'L!(!) , #u $ D(") ,
!a is a distribution of order 0 in ", called the Dirac mass at point a. However, !a cannot be obtainedvia a L1
loc(") function. Indeed, suppose that there exists f $ L1loc(") such that
)!a, u+ ="
!f(x)u(x) dx = u(a) , #u $ D(") . (1.9)
By denoting " = "\{a}, we have
)!a, u+ ="
!f(x)u(x) dx = 0 , #u $ D(") .
Hence, we can show that f = 0 almost everywhere in " and thus almost everywhere in ". Therefore,(! f(x)u(x) dx = 0 for every function u $ D("). This is in contradiction with he assumption (1.9)
when u(a) "= 0.
The continuity property of a distribution can be characterized using sequences, as for the linear operatorsin normed spaces. We introduce first the notion of convergence in D(").
Definition 1.8 Let consider an open set " & Rd. The sequence (un)n'N of elements of D(") convergestoward u $ D(") if
1. there exists a compact set K in " such that Supp (un) & K, for all n $ N.
2. for every # $ Nn, we have limn%#
'"!(un % u)'# = 0.
Definition 1.9 Let consider a sequence (Tn)n'N of elements in D!(") and a distribution T on ". Thesequence (Tn)n'N is said to converge toward T if and only if:
#u $ D(") , limn%#
)Tn, u+ = )T, u+ .
Theorem 1.9 A linear form T : D(") ( C is a distribution on " if and only if, for any sequence(un)n'N & D(") tending toward 0 in D("), we have:
limn%#
)T, un+ = 0 .
12 Chapter 1. Distributions on open sets of Rd
Proof. Suppose T $ D$(") and consider (un)n"N & D(") tending toward 0 in D("). From the previous definitionof the convergence in D("), we know that there exists a compact subset K in " such that Supp (un) & K for eachn $ N. Hence, there exist constants CK and NK such that
|)T, un+| - CK
#
|"|%NK
'""un'! , #n $ N .
Since limn&! '""un'! = 0 for each given #, we deduce the result for every sequence (un)n"N & D(") tendingtoward 0 in D("). The reciprocal assertion is established by contradiction.
Suppose there exists a compact K0 in " such that for all N $ N and for every C > 0 we could find uN,C $DK0(") verifying
|)T, uN,C+| > C#
|"|%N
'""uN,C'! .
Take uN = uN,N for every N $ N'. It is obvious that uN "= 0, hence assume that the following equality holds
sup|"|%N
'""uN'! =1N
.
Hence, the two previous equalitites yield the result
|)T, uN +| > 1 , #N $ N .
Since, for every N $ N, Supp (uN ) & K0 we deduce
limN&!
uN = 0 , in D(") ,
that is obviously in contradiction with the hypothesis. !We may be wondering what is the meaning of the integral
(! f(x)u(x) dx for arbitrary functions and
for every test function u. Obviously, the function f must be in L1(K) for every compact set K in ". Thefollowing theorem allows to consider functions in L1
loc(") as distributions.
Theorem 1.10 Consider a test function u $ DK("). The linear map + : L1loc(") ( D!("), f /( +(f) :
u ((! f(x)u(x) dx is a linear injection. Furthermore, for any compact K in ", we have
#u $ DK(") , |)+(f), u+| - 'f'L1(K)'u'L!(!) .
Remark 1.1 We will assume now that any function f in L1loc(") is identical to the distribution
u /("
!f(x)u(x) dx .
When considering that the distribution T is a function, this will mean that there exists a function f $L1
loc(") such that
#u $ D(") , )T, u+ ="
!f(x)u(x) dx .
The following proposition establishes that the Dirac mass is the limit of the approximations of identity.
Proposition 1.5 Consider a function , in D(Rd) of integral 1 and ('n)n'N a sequence of real numberstending toward 0. We pose
,#n(x) def= '"dn ,
&x
'n
'.
Then, the sequence (,#n)n'N converges toward !0 in the distributional sense, where !0 is the linear formdefined by !0 : D(Rd) ( C, u /( u(0).
1.2. The concept of distribution 13
Remark 1.2 This result has a physical meaning. Indeed, if the functions 'n are considered as massdensities, the limit (in a certain sense) must be seen as a ponctual mass associated with the origin.
Definition 1.10 We call principal value of the function 1/x and we denote it pv1x
the distributiondefined by:
)pv1x
, u+ =12
" #
"#
u(x)% u(%x)x
dx , #u $ D(R) .
Notice that this formula defines a distribution since we have, for every test function u $ D["R,R],$$$$)pv
1x
, u+$$$$ - R'u!'L! .
1.2.2 Operations on distributions
We define now several usual operations on distributions that are already familiar for classical smoothfunctions, for example test functions.
Definition 1.11 (Restriction) Suppose - and " are two open subsets in Rd such that - & " andconsider a distribution T on ". We define the restriction of T to the open set -, denoted by T|%, as
#u $ D(-) , )T|%, u+ def= )T, u+ .
Furthermore, if f $ L1loc(") we have:
#u $ D(") ,
"
%f|%(x)u(x) dx =
"
!f|%(x)u(x) dx .
This definition coincide with the notion of restriction for functions.
Remark 1.3 The restriction of a distribution to a set is only defined for an open set.
If T1 and T2 are distributions of order m on Rd, it is obvious that T1 + T2 and aT1 are distributions ofthe same order for all a $ R. However, the product T1T2 of two distributions is not a distribution. Forexample, if we were to define )T1T2, u+ = )T1, u+)T2, u+, this is not linear in u.
1.2.3 Di!erentiation of distributions
One of the strongest advantage of the distribution theory is that the di!erentiation operation is alwaysdefined and is ”continuous”. Indeed, consider a function f $ C1(Rd) and u $ D(Rd), then:
6"f
"xi, u
7=
"
Rd
"f
"xi(x)u(x) dx ,
= %"
Rd
"u
"xi(x)f(x) dx = %
6f,
"u
"xi
7.
This result is still valid if we set f to be a distribution. More precisely,
Proposition 1.6 Let T $ D!(") be a distribution on an open set " & Rd. The linear form T (j) on D(")defined by:
)T (j), u+ def= %6
T,"u
"xj
7= %)T, "xju+ ,
defines a distribution on ". Furthermore, if (Tn)n'N is a sequence in D!(") converging toward T , thenthe sequence (T (j)
n )n'N converges toward T (j).
14 Chapter 1. Distributions on open sets of Rd
Proof. Since T is a distribution, for every compact K in ", there exists a constant C and an integer N suchthat
#v $ DK , |)T, v+| - C sup|"|%N
'""v'L! .
For every u $ DK , we apply this to v = "xj u $ DK . Hence, we have
#u $ DK , |)T (j), u+| = |)T, "xj u+| - C sup|"|%N+1
'""u'L! .
Thus, T (j) is a distribution on ". Moreover, for every function u $ D(") we have:
limn&!
)T (j)n , u+ = % lim
n&!)Tn, "xj u+ = %)T, "xj u+ = )T (j), u+ .
the results follows. !
Definition 1.12 Suppose T $ D!("). Then, the distribution T (j) is called the partial derivative of Twith respect to the variable xj and is denoted "xjT and is such that:
)"xjT, u+ = %)T, "xju+ , #u $ D(") . (1.10)
Notice that if f is a di!erentiable function, the derivative of the distribution associated with f coincidewith the usual derivative of f .
Example 1.2 (i) The Heaviside step function H is the characteristic function of R+. We have "xH =!0 in D!(R). Indeed, for any test function u $ D(R), we have:
)"xH,u+ =6
dH
dx, u
7= %)H,u!+ = %
" +#
"#u!(x)H(x) dx
= %" +#
0u!(x) dx = u(0) since u(+!) = 0
= )!0, u+ .
(1.11)
Hence, the derivative of H in the sens of distributions is the Dirac mass defined hereabove.
(ii) Since the function f(x) = log |x| is in the space L1loc(R), it defines an element of D!(R). Indeed,
the derivative of f in the distributional sense is the distribution pv1x
introduced previously:
df
dx= pv
1x
.
According to the definition of the derivative in D!(R), we have for all u $ D(R):6
df
dx, u
7= %
"
Rlog(|x|)u!(x) dx .
We observe that:"
|x|>#log(|x|)u!(x) dx = [u(%')% u(')] log '%
"
|x|>#u!(x) dx ,
and thus, we deduce that"
Rlog(|x|)u!(x) dx = lim
n%#
"
|x|>#log(|x|)u!(x) dx = %)pv
1x
, u+ .
1.2. The concept of distribution 15
The next results establishes a relation between the derivative and the primitive, in the distributionalsense.
Theorem 1.11 Consider a distribution T on an open interval I & R. If its derivative, in the distribu-tional sense, T ! is a function of L1
loc(I), then T is a continuous function and we have, for all a $ I:
T (x) =" x
aT !(x)dx + C .
Lemma 1.6 Suppose I is an open interval of R. Then,
1. the distributions on I for which the identity T ! = 0 holds, are the constant functions.
2. for any T2 $ D!(I), there exists T1 $ D!(I) such that T !1 = T2.
3. for a $ I and f $ L1loc(I), we have (in the distributional sense):
d
dx
" x
af(y)dy = f(x) .
Proof. Let consider % $ D(I), such that"
I%(x) dx = 1, and u $ D(I). Then, the function
((x) = u(x)%&"
Iu(x) dx
'%(x) $ D(I) and
"
I((x) dx = 0 .
This implies the existence of a unique function v $ D(I) such that v$ = (. Hence, there exists a unique functionv $ D(I) such that
v$ = u%&"
Iu(x) dx
'% . (1.12)
Now, let consider T $ D$(I) such that T $ = 0. We have
)T, u+ =&"
Iu(x) dx
')T, %++ )T, v$+ .
The term )T, v$+ = %)T $, v+ is equal to zero and we obtain by denoting C the constant term )T, %+:
)T, u+ = C
"
Iu(x) dx .
Thus, T is equal to a constant C.To find a primitive T1 of T2, we pose
)T1, u+ = %)T2, v+ ,
where v is the unique function of D(I) associated with u via the relation (1.12). It is easy to show the linearityand continuity of T1 and thus we have
)T1, u$+ = %)T2, u+ , #u $ D(I) ,
and thus we have proved that T $1 = T2. !
Lemma 1.7 Consider T $ D!(Rd) and suppose that "xjT = 0 for all i = 1, . . . , d. Then, T is a constant(function).
16 Chapter 1. Distributions on open sets of Rd
1.2.4 Multiplication by a C# function
Proposition 1.7 Let consider f $ C#(") and T $ D!("). Then, for all u $ D("), the linear form onD(") defined by u /( )T, fu+ is a distribution on ". The order of this distribution on every compact Kin " in lesser than or equal to the order of T on K.
Definition 1.13 The product of the distribution T $ D!(") by the function f $ C#(") is the distributiondefined by:
)fT, u+ = )T, fu+ , #u $ D(") .
Example 1.3 (i) For f $ C#(") and a $ ", we have f!a = f(a)!a. Indeed, we can observe that
)f!a, u+ = )!a, fu+ = )f(a)!a, u+ , #u $ D(") .
In particular, x!0 = 0.
(ii) We can find all solutions T $ D!(") of the equation xT = 0. Let consider , $ D(R) such that,(0) = 1. For every function u $ D(R), the function u% u(0), vanishes at the origin, hence thereexists a function v $ C#(R) such that
u = u(0), + xv .
Since u and , are functions with compact support, the relation means that v $ D(R). Thus, wehave
)T, u+ = u(0))T, ,++ )T, xv+ .
By assumption, we have )T, xv+ = )xT, v+ = 0. Denoting C the constant )T, ,+, we obtain that)T, u+ = Cu(0), hence T = C!0.
Proposition 1.8 (Leibniz formula) Consider f $ C#(") and T $ D!("). Then,
"xi(fT ) ="f
"xiT + f"xiT , #i $ {1, . . . , n} .