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334 6. Introduction to Pseudodifferential Operators

6.2.3 Levi Functions for Scalar Elliptic Equations

From the homogeneous constant coefficient case we have an explicit formula
to construct the fundamental solution via the Fourier transformation. How-
ever, in the case of variable coefficients, fundamental solutions can not be
constructed explicitly, in general. This leads us to the idea of freezing coeffi-
cients in the principal part of the differential operator and, by following the
idea of the parametrix construction, one constructs the Levi functions which
dates back to the work of E.E. Levi [187] and Hilbert [125]. These can be
used as approximations of the fundamental solution for the variable coeffi-
cient equations. In general, one can show that Levi functions always exist
even if there is no fundamental solution.
Pomp in [250] developed an iterative scheme for constructing Levi func-
tions of arbitrary order for general elliptic systems, from the distributional
point of view. Here we combine his approach with the concept of pseudodif-
ferential operators.
To illustrate the idea, we use the scalar elliptic equation of the form (3.6.1)
with p = 1 and the order 2κ (instead of 2m). For fixed y ∈ Ω, we define

α
P 0 (y)v(x)= a α (y)D v(x) (6.2.29)
x
|α|=2κ
and write the original equation in the form

Pu(x)= P 0 (y)u(x) − T(y)u(x)= f(x) , (6.2.30)

where

T(y)u(x)= P 0 (y) − P u(x) (6.2.31)

α α
= a α (y) − a α (x) D u(x) − a α (x)D u(x) .
x x
|α|=2κ |α|<2κ
Note that P 0 (y) is the principal part of P with coefficients frozen at y.
Its fundamental solution exists and has the form
e
i(x−z)·ξ
−n
E 0 (x, z; y)=(2π) 0 dξ (6.2.32)
a (y, ξ)
2κ
IR n
with a parameter y, where
0
a (y, ξ)=(−1) κ a α (y)ξ α (6.2.33)
2κ
|α|=2κ
is the (principal) symbol of P 0 (y). If one identifies z = y, then L 0 (x, y)=
E 0 (x, y; y) is called the Levi function of order 0 for P in (6.2.30) and has the
singular behaviour of the form