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An Introduction to Partial Differential Equations by Jacob Rubinstein, Yehuda Pinchover

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9

Equations in high dimensions

9.1 Introduction

To simplify the presentation we have concentrated so far mainly on equations involving two independent variables. In this chapter we shall extend the discussion to equations in higher dimensions. A considerable part of the theoretical and practical aspects that we studied for equations in two variables can be extended at once to higher dimensions. Nevertheless, we shall see that there are sometimes significant differences between problems in different dimensions.

9.2 First-order equations

The general first-order quasilinear equation for a function u in n variables is

image

(9.1)

The method of ...

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