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## The Symbol of an Operator

The operator discussed above does not correspond to any differential operator. Only polynomials in correspond to differential operators, and this is via a very simple relation. The differential operator corresponds to the symbol since

 (22)

From this we have,
 (23)

where and .

For a general operator , defined on functions in an infinite space, we define the symbol by

 (24)

This definition is for scalar PDE as well as for systems of PDE. In the second case the symbol will be a matrix whose elements are functions of . A definition in a bounded domain can also be done, by considering a small vicinity of a point in and a localization of the above.

Using the definition (24) we see that we can define a larger class of operators by considering symbols which are not polynomials. Under some assumptions which we omit here, one gets the class of pseudo-differential operators. They play a very important role in the study of boundary value problems for elliptic equations.

Example IV. We consider next the example

 (25)

where is any of the two domains in the previous example. Following the same procedure as before,
 (26)

 (27)

implies
 (28)

and the relevant solution is
 (29)

The value of is determined from the boundary condition and clearly we have
 (30)

The relation of to the boundary values is described by a mapping
 (31)

whose symbol can be easily found by differentiating in the outward normal direction at the boundary, giving
 (32)

Notice that although we are dealing with differential problems, some relation between boundary values are not governed anymore by differential operators. The operators from the last two examples are therefore not differential operators but pseudo-differential operators.

Shape design problems are related to boundary control problems, and therefore these type of operators play a very important role in shape optimization. They will help us to characterize the minimization problem in quantitative way that will allow the construction of very effective solvers.

Next: Fourier Analysis For Optimization Up: On Pseudo-Differential Operators Previous: On Pseudo-Differential Operators
Shlomo Ta'asan 2001-08-22