2.3 WEAK FORMULATIONS 41
the description of the (physical) field as a connector between geometrical
entities.
Which somewhat devalues differential operators, too: grad, rot and
div, in this light, appear as auxiliaries in the expression of conservation
relations, as expressed by the Ostrogradskii and Stokes theorems. Their
failure to make sense locally is thus not to be taken too seriously.
Proper form is given to the foregoing ideas in
differential geometry.
There, one forgets about the scalar or vector fields and one focuses on the
mappings they represent (and thus, to some extent, hide). Fields of
linear mappings of type
GEOMETRIC_ELEMENT --~ REAL
are called
differential forms,
of
degree 0
to 3 according to the dimension of the
geometric objects they act upon, and under regularity assumptions which
are milder than for the scalar or vector proxies, one defines a unique
operator d, the
exterior differential,
which is realized as grad, rot, or div,
depending on the dimension.
All
laws of electromagnetism can be cast
in this language (including constitutive laws, which are mappings from
p-forms to (3 - p)-forms, with p = 0 to 3).
The moderate approach we now follow does not go so far, and keeps
the fields as basic objects, but stretches the meaning of the differential
operators, so that they continue to make sense for some discontinuous
fields. The main idea is borrowed from the theory of distributions: Instead
of seeing fields as collections of pointwise values, we consider how they
act on other fields, by integration. But the full power of the theory of
distributions is not required, and we may eschew most of its difficulties.
2.3 WEAK FORMULATIONS
First, some notation. Symbols
C k
and C a for smoothness have already
been introduced, compact support 9 is usually denoted by a subscripted
0, and blackboard capitals are used in this book to stress the vector vs
scalar opposition when referring to spaces of fields. Putting all these
conventions together, we shall thus have the following list of infinite-
dimensional linear spaces:
• ck(E3 ) • The vector space of all k-smooth functions in E3,
• ok(E3 )" All k-smooth vector fields in E3,
9The
support
of a function, real- or vector-valued, is the closure of the set of points
where it doesn't vanish. Cf. A.2.3.
42 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
• c0k(D),
~0 k(D)"
Same, with compact support contained in D,
being understood that domain D can be all E3, and finally,
• ck(w ), ck (D),
for the vector spaces of restrictions to D (the closure
of D), of k-smooth functions or vector fields. In all
of these, k can be replaced by ~o. (In the_ inset, the
supports of a q)l E C0~(D) and a q)2 E C ~(D), which is
thus the restriction of some function defined beyond
D, whose support is sketched.)
" W
2.3.1 The "divergence" side
Now, let's establish a technical result, which generalizes integration by
parts. Let D be a regular domain (not necessarily bounded), S its
boundary, b a smooth vector field, and q0 a smooth function, both with
compact support in
E 3
(but their supports may extend beyond D).
Form u = q0 b. Ostrogradskii's theorem asserts that SD div u = Js n. u,
with n pointing outwards, as usual. On the other hand, we have this
vector analysis formula,
div(q0 b) = q~ div b + b. grad q0.
Both things together give
(9)
SD q0
div b =
-
~D b. grad q0 + Ss n. b q),
a fundamental formula.
By (9), we see that a 1-smooth divergence-free field b in D is
characterized by
(10)
S D b.
grad q) - 0 V q0 ~ C01(D),
since with q0 = 0 on the boundary, there is no boundary term in (9). But
(10) makes sense for fields b which are only
piecewise
smooth. 1° We
now take a bold step:
Definition
2.1.
A piecewise smooth field b which satisfies
(10)
will be said to
be
divergence-free,
or solenoidal,
in the weak sense.
The q)'s in (10) are called
test functions.
1°All that is required is the integrability of b. grad q0 in (10), so 0-smoothness, that is,
continuity, of each "piece" of b is enough.
2.3 WEAK FORMULATIONS 43
A solenoidal smooth field is of course weakly divergence-free. But
from our earlier discussion, we know that the physical b, in magnetostatics,
is only piecewise smooth, and satisfies transmission conditions. Hence
the interest of the following result:
Proposition 2.1.
For a piecewise smooth
b, (10)
is equivalent to the statement
"div b = 0
inside regularity regions and
[n. b] = 0 at their interfaces".
Proof.
Recall that "piecewise" means that D can be partitioned into a
,finite
number of subdomains D i in which b is smooth, so a proof with
two
subdomains will be enough. It's a long proof, which will require two
steps.
We begin with the shorter one, in which
b is supposed to be solenoidal in D 1 and D 2
separately (notation in inset), with In. b] = 0
on the interface Z. Our aim is to prove (10).
Let q0 be a test function. Formula (9) holds
in D 1 and D 2 separately, and gives
//
~n
// ~ 1 ]
/ / /
/ D ]
/ D1 ~ 2 /
i n / /
/
fD b. grad q0-
~D1 b .
grad q0 + ~D2 b. grad q~
=-JD1 q0 div b + Jr n l. b q~- ~D2 q~ div b + Jz
ha"
= ~z [n. b] q0 = 0
bq0
since [n. b] = 0 has been assumed, hence (10). The absence of surface
terms on S is due to the inclusion supp(q0) c D.
Conversely, suppose (10) holds. Since (10) says
"for all
test functions",
let's pick one, % which is supported in
D1,
and apply (9). There is no
surface term, since supp(q0) does not intersect Z, so 0 =j~ div b q0.
(Note that div b is a smooth function there.) This holds for
all cp
C01(D1). 11 But the only way this can happen (see A.2.3 if this argument
doesn't sound obvious) is by having div b = 0 in
D 1.
Same reasoning in
D2,
leading to div b = 0 also in
D 2. Now,
start from (10) again, with a
test function which does not necessarily vanish on Z, and use the newly
acquired knowledge that div b = 0 in D 1 and DR:
0--I D b. grad q0- fD 1 b. grad q)+ fD 2 b. grad q0 =
----- ~ n 1 .
b q0 + jz n 2. b q) = Jz [n. b] q~
for all q0 e C01(D). But such test functions can assume any value on Z, so
11Which is why the presence of the quantifier V in Eq. (10) is mandatory. Without it,
the meaning of the statement would change totally.
44 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
again, the only way ~ [n. b] q0 can vanish for all of them is by [n. b]~
being 0.
Interface conditions are thus implicit
in the "weak solenoidality" condition
(10). We shall therefore acquire the "weak formulation reflex": Each
time a statement of the form "div b = 0" appears in the formulation of a
problem (this is what one calls the "strong formulation"), replace it by
the weak formulation (10). This does no harm, since there is equivalence
in case b has a divergence in the ordinary ("strong") sense. It does
some good if b is only piecewise smooth, since there is no need to make
explicit, or even mention, the transmission conditions In. b ] = 0, which
are implied by (10), as Prop. 2.1 has shown.
We now see why the q0's are called "test functions": By carefully
selecting them, we were able to "test" the equality div b = 0 inside
regularity regions, to "test" the transmission condition over Z, etc. The
function div b and the constant 0 were thus deemed equal not because
their
values
would coincide, but on the ground that their
effects
on test
functions were the same. (This principle, duly abstracted, was the foun-
dation of the theory of distributions.)
Remark 2.1. The reader aware of the "virtual work principle" in mechanics
will have recognized the analogy: There too, fields of forces are tested
for equality by dot-multiplying them by fields of virtual displacements
and integrating, and two force fields are equal if their virtual works
always coincide.
An obvious generalization of (10) is
(11) -~Db.gradq0--~Dfq0 Vq0~ C01(D),
where f is a given function (piecewise smooth). This means "div b = f
in the weak sense". (Exercise 2.2: Check that.)
Remark 2.2. One may wonder to which extent weak solenoidality depends
on the regularity of test functions, and this is a good question, since of
course the following statement, for instance,
(10')
YD b.
grad q0 - 0 V q0 ~ C0~(D),
is logically
weaker
than (10): There are fewer test functions, hence fewer
constraints imposed on b, and hence, conceivably, more weakly solenoidal
fields in the sense of (10') than in the sense of (10). The notion would be
of dubious value if things went that way. But fortunately (10)
and
(10')
are equivalent:
This results from the density property proved in Section
A.2.3: Given a Co I test function q0, there exists a sequence {q0n} of
2.3 WEAK FORMULATIONS
45
C0~(D) functions such
that ;D
I grad(q°n- q))]2 ~
0.
Then (Exercise 2.3:
Check it)
~D b .
grad q0 = lim n -,
~
;D b . grad q)n if (10') holds, which
implies (10). ~)
Remark 2.3. As a corollary of Remark 2.2, any function q0 such that
grad q0 is the limit, in the above sense, of a sequence of gradients of test
functions, also qualifies as a test function. We shall remember this in due
time.
Remark 2.4. If the superscript k in c0k(D) is thus not crucial (all that is
required of % in terms of regularity, is to have a square-integrable gradient),
what about the subscript 0, denoting compact support ? That is essential.
If test functions could assume nonzero values on the boundary, this would
put more constraints on b than mere solenoidality. We shall make use of
this too, when treating boundary conditions.
2.3.2 The "curl" side
All of this cries out for symmetrization: What we did with the divergence
operator should have counterparts with the curl operator. This time we
know the way and will go faster.
Let a and h both belong
to
C01(E3 ), and let D be as above. Form
u = h x a. We have this other vector analysis formula,
div(h x a) = a. rot h- h. rot a,
and by Ostrogradskii again, we get
(12)
JD h . rot
a
=
fD a. rot h - Js n x h. a,
the second fundamental integration by parts formula, on a par with (9).
Remark 2.5. Note the formal analogies, and also the differences, between
(9) and (12): grad became rot,-div became rot too, x replaced
the dot, signs changed in puzzling patterns ... Obviously these two
formulas are, in some half-veiled way, realizations of a unique one, which
would call for different notation and concepts: those of differential
geometry. ~)
By (12), a smooth curl-free field h in D is characterized by
(13)
~D h.
rot a - 0 V a ~ C01(D).
Again (13) makes sense for non-smooth fields h, if they are square-
integrable, and hence (now the obvious thing to do):
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