Dorina Mitrea - Distributions, Partial Differential Equations, and Harmonic Analysis

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Dorina Mitrea Universitext Distributions, Partial Differential Equations, and Harmonic Analysis 2013 10.1007/978-1-4614-8208-6_1 Springer Science+Business Media New York 2013

1. Weak Derivatives

Dorina Mitrea 1

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

Abstract

Starting from the discussion of the Cauchy problem for a vibrating infinite string as a motivational example, the notion of weak derivative is introduced as a mean of extending the notion of solution to a more general setting, where the functions involved may lack standard pointwise differentiability properties. Here two classes of test functions are also defined and discussed.

1.1 The Cauchy Problem for a Vibrating Infinite String

The partial differential equation

(1.1.1)

was derived by Jean dAlembert in 1747 to describe the displacement u ( x 1, x 2) of a violin string as a function of time and distance along the string. Assuming that the string is infinite and that at time x 2=0 the displacement is given by some function leads to the following global Cauchy problem

(1.1.2)

Thanks to the regularity assumption on , it may be checked without difficulty that the function

(1.1.3)

is a solution of () would continue to satisfy .

To answer this question, fix a function satisfying pointwise in . If is an arbitrary function and R (0,) is a number such that , then integration by parts gives

(1.1.4)

Note that the condition for all is meaningful even if , which suggests the following definition.

Definition 1.1.

A function is called a weak ( generalized ) solution of the equation in if

(1.1.5)

Returning to () is a generalized solution of in . Concretely, fix and write

(1.1.6)

where for the last equality in () we have made the change of variables , . If we now let for , then

(1.1.7)

and

(1.1.8)

which, when used in (), give

(1.1.9)

Let R (0,) be such that . Then the support of is contained in the set of points satisfying

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