# 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 functional**

**analysis 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**

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