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Measure Theory

Vector Space

Calculus of Variation

Partial Differential Euqations

Introduction

Elementary Methods

Complex Variable

Miscellaneous

Haoyu Tang

Definition

Definition 1

A PDE for a function $u(x_1,...,x_n)$ is of the form:

F(x_1,...,x_n, u,u_{x_1},...,u_{x_n},u_{x_1x_1},u_{x_1x_2},...)=0

The order is the order of the highest-order derivative
An equation is called linear if it is linear on $u$ and its derivatives
An equation is callled semi-linear if all coefficinets are only functions of $x_1,...,x_n$
An equation is called quasilinear if coefficinet $a_N$ is only functions of $x,u$ and lower-order derivatives.