2 1. INTRODUCTION

is given and

u : U → R

is the unknown.

We solve the PDE if we ﬁnd all u verifying (1), possibly only among those

functions satisfying certain auxiliary boundary conditions on some part Γ

of ∂U. By ﬁnding the solutions we mean, ideally, obtaining simple, explicit

solutions, or, failing that, deducing the existence and other properties of

solutions.

DEFINITIONS.

(i) The partial diﬀerential equation (1) is called linear if it has the form

|α|≤k

aα(x)Dαu

= f(x)

for given functions aα (|α| ≤ k), f. This linear PDE is homogeneous

if f ≡ 0.

(ii) The PDE (1) is semilinear if it has the form

|α|=k

aα(x)Dαu

+

a0(Dk−1u,

. . . , Du, u, x) = 0.

(iii) The PDE (1) is quasilinear if it has the form

|α|=k

aα(Dk−1u,

. . . , Du, u,

x)Dαu

+

a0(Dk−1u,

. . . , Du, u, x) = 0.

(iv) The PDE (1) is fully nonlinear if it depends nonlinearly upon the

highest order derivatives.

A system of partial diﬀerential equations is, informally speaking, a col-

lection of several PDE for several unknown functions.

DEFINITION. An expression of the form

(2)

F(Dku(x), Dk−1u(x),

. . . , Du(x), u(x), x) = 0 (x ∈ U)

is called a

kth-order

system of partial diﬀerential equations, where

F :

Rmnk

×

Rmnk−1

× · · · ×

Rmn

×

Rm

× U →

Rm

is given and

u : U →

Rm,

u =

(u1,

. . . ,

um)

is the unknown.

Here we are supposing that the system comprises the same number m

of scalar equations as unknowns

(u1,

. . . ,

um).

This is the most common

circumstance, although other systems may have fewer or more equations

than unknowns. Systems are classiﬁed in the obvious way as being linear,

semilinear, etc.