Computational Electromagnetism

A.4 GLIMPSES OF FUNCTIONAL ANALYSIS

317

orthocomplement of dom(A) with respect to X, and hence a nontrivial

pair {x, 0} ~ A I. ~)

Remark A.11. After (11), the domain of A* is made of all y such that

the linear partial function x --~ (y, Ax)y be continuous on dora(A), with

respect to the metric of X. This can be used as an alternative definition

of A*: first define its domain this way, then define the image A*y as

the Riesz vector of the linear continuous mapping obtained by extending

x --~ (y, Ax)y to the closure of dora(A), i.e., all X, by continuity.

If dora(A) is not dense, we can always consider A as being of type

X ° ~ Y, where X ° is the closure of dora(A) (equipped with the same

scalar product as X, by restriction), and still be able to define an adjoint,

now of type Y --~ X °.

Note that (A±) ± is the closure of A. Therefore, if an operator A has

an adjoint, and if dora(A*) is dense, the closure of A is A**, the adjoint

of its adjoint. Therefore,

Proposition A.2.

Let A : X --~ Y be a linear operator with dense domain. If

dom(A*)

is dense in

Y, A

is closable.

Its closure is then A**. This is how we proved that div was closable, in

Chapter 5: The domain of its adjoint is dense because it includes all

functions q0 ~ C0°°(D). Indeed, the map b

-o YD q)div

b =- YD b. grad q0

• 2 •

is IL-continuous for such a q0, due to the absence of a boundary term.

As we see here, the weak divergence is simply the adjoint of the operator

2 2

grad" C0°°(D) --~ C0°°(D), the closure of which in L (D) x IL (D), in turn, is

a strict

restriction (beware!) of the weak gradient.

The reader is invited to play with these notions, and to l~rove what

follows: The boundary of D being partitioned S = S h u S ~ as in the

main chapters, start from grad and - div, acting on smooth fields, but

restricted to functions which vanish on S h and to fields which vanish on

S b, respectively. Show that their closures (that one may then denote

grad h and- div b) are mutual adjoints. Same thing with rot h and rot b.

REFERENCES

[AB]

[Bo]

[B1]

[B2I

Y. Aharonov, D. Bohm: "Significance of Electromagnetic Potentials in the Quantum

Theory", Phys. Rev., 115 (1959), pp. 485-491.

R.P. Boas: "Can we make mathematics intelligible?", Amer. Math. Monthly, 88

(1981), pp. 727-731.

A. Bossavit: "The Exploitation of Geometrical Symmetry in 3-D Eddy-currents

Computations", IEEE Trans., MAG-21, 6 (1985), pp. 2307-2309.

A. Bossavit: "Boundary value problems with symmetry, and their approximation

by finite elements", SIAM J. Appl. Math., 53,5 (1993), pp. 1352-1380.