The Symbol of an Operator

The operator $T$ discussed above does not correspond to any differential operator. Only polynomials in ${\bf k}$ correspond to differential operators, and this is via a very simple relation. The differential operator $\partial/\partial x_j$ corresponds to the symbol $(i k_j)$ since

$$\frac{\partial}{\partial x_j} \exp ( i {\bf k} \cdot {\bf x} ) = i k_j \exp ( i {\bf k} \cdot {\bf x} ).$$ (22)

From this we have,

$$\mbox{\tt Symbol: } \sum _{\vert\gamma \vert \leq m} a_\gamma (i ... ...\tt Differential Operator: } \sum _{\vert\gamma \vert \leq m} a_\gamma D^\gamma$$ (23)

where $i {\bf k} = (i k_1, \dots, i k_n ), \gamma = (\gamma _1, \dots, \gamma_n), (i{\bf k})^\gamma = (i k_1)^{\gamma _1} \dots (i k_n)^{\gamma _n}$ and $D^\gamma = (\partial ^{\gamma _1}/\partial x_1^{\gamma _1})\dots(\partial ^{\gamma _n}/\partial x_n^{\gamma _n})$.

For a general operator $T$, defined on functions in an infinite space, we define the symbol $\hat T({\bf k})$ by

$$T \exp ( i {\bf k} \cdot {\bf x} ) = \hat T({\bf k}) \exp ( i {\bf k} \cdot {\bf x} ).$$ (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 ${\bf k}$. A definition in a bounded domain $\Omega$ can also be done, by considering a small vicinity of a point in $\Omega$ 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

$$\begin{array}{ll} \Delta \phi = 0 & \Omega \\ \ \phi = \alpha & \partial \Omega \end{array}$$ (25)

where $\Omega$ is any of the two domains in the previous example. Following the same procedure as before,

$$\alpha({\bf x}) = \exp ( i {\bf k} \cdot {\bf x} )$$ (26)

$$\phi({\bf x} , z ) = A \exp ( i {\bf k} \cdot {\bf x} ) \exp ( \sigma z )$$ (27)

implies

$$\sigma ^2 - \vert {\bf k} \vert ^2 = 0$$ (28)

and the relevant solution is

$$\phi ({\bf x}, z) = A \exp ( i {\bf k} \cdot {\bf x} ) \exp ( - \vert{\bf k}\vert z ).$$ (29)

The value of $A$ is determined from the boundary condition and clearly we have

$$A = 1.$$ (30)

The relation of $\frac{\partial \phi}{\partial n}_{\vert _{\partial\Omega}}$ to the boundary values $\phi _{\vert _{\partial\Omega}}$ is described by a mapping $S$

$${\frac{\partial \phi}{\partial n}}_{\vert _{\partial\Omega}}= S \alpha$$ (31)

whose symbol can be easily found by differentiating $\phi$ in the outward normal direction at the boundary, giving

$$\hat S ({\bf k} ) = \vert {\bf k}\vert.$$ (32)

Notice that although we are dealing with differential problems, some relation between boundary values are not governed anymore by differential operators. The operators $S,T$ 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.