# Linear Partial  Differential Equations

The most general linear partial differential operator of order m is of
the form
A = P(x, D) = X
|µ|≤m
aµ(x)Dµ
,
where the coefficients aµ(x) are functions of x.
The usual problem one encounters concerning linear partial differential equation is as follows: we are given a continuous function f and a partial differential operator A of order m with continuous coefficients. We search for a function u having continuous derivatives
of order m such that
Au = f.
We would like to know the set of functions f for which there is a solution and how many solutions there are. Expressing everything in terms of continuity has led to very few results. All that is available to us is the maximum norm, and very little is achieved by studying the problem only using continuity.
On the other hand, the use of other methods of functional analysis has led to very fruitful results. Then we have a choice of spaces and useful properties. The spaces that have the most flexible properties and produce the strongest result appear to be the L
p
spaces. Once these have been chosen, one must decide what spaces should be used for the given function f. One must consider the operator A acting on that space. Then one realizes that there are a multitude of theorems
which can hold for the problem at hand. Hypotheses will depend on the makeup of the operator A. In this volume we shall show the types of results that one can obtain depending on the operator.
Once a space X has been selected, an important question to ask
is if there exists a number λ such that the equation
(A − λ)u = f
has a unique solution for each f ∈ X. If not, this may be an indication that we are picking the wrong space. If such a number exists, this indicates that we are in the right space. In this case, we say that λ
is in the resolvent set ρ(A) of A.
One useful concept is that of the spectrum of an operator. By definition, a scalar λ is in the spectrum of a closed operator A if
A − λ does not have a bounded inverse, i.e., λ is not in the resolvent
set ρ(A) of A. It is important to know the spectrum of A in order to
determine the functions f ∈ X for which the equation (A − λ)u = f
has a solution.
If λ is not in the resolvent set of A, then we know that we cannot
solve the equation (A − λ)u = f for every f ∈ X. We would like
to know the set of functions for which we can solve. Here functionalanalysis helps out. There may be a collection of numbers λ which are not in the resolvent set ρ(A) but are almost in the resolvent. We say that they are in the spectrum, but not in the essential spectrum
σe(A), the hardcore spectrum of A (see Chapter 1). In this case we can characterize the set of functions for which we can solve. An appropriate theorem states