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.
