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=
be the set of all partial derivatives of order
k
.If
k
1, we regard the elements of
D
1
u(x)
=:
Du(x)
as being arranged in a row vector
=
Du
(∂
x
1
u,...,∂
x
d
u).
2, we regard the elements of
D
2
u(x)
as being arranged in a matrix
If
k
=
⎛
⎝
⎞
⎠
∂
x
1
∂
x
1
u
... ∂
x
1
∂
x
d
u
.
.
.
.
.
D
2
u
=
.
∂
x
d
∂
x
1
u
...
∂
x
d
∂
x
d
u
Hence, the Laplacian
u
of
u
can be written as
d
D
2
u
u
:=
∂
x
i
∂
x
i
u
=
tr
[
]
,
(2.2)
i
=
1
]=
i
=
1
B
ii
is the trace of a
d
d
×
d
: R
→ R
→
[
×
where tr
d
-matrix
B
.In
the following, we write
∂
x
i
x
j
instead of
∂
x
i
∂
x
j
to simplify the notation.
A
partial differential equation
is an equation involving an unknown function
of two or more variables and certain of its derivatives. Let
G
,
B
tr
B
d
⊂ R
be open,
x
=
(x
1
,...,x
d
)
∈
G
, and
k
∈ N
.
Definition 2.2.1
An expression of the form
F(D
k
u(x), D
k
−
1
u(x),...,Du(x),u(x),x)
=
0
,x
∈
G,
is called a
k
th order partial differential equation, where the function
d
k
d
k
−
1
d
F
: R
× R
×···×R
× R ×
G
→ R
is given and the function
u
:
G
→ R
is the unknown.
Let
a
ij
(x), b
i
(x), c(x)
and
f(x)
be given functions. For a
linear first order PDE
in
d
+
1 variables,
F
has the form
d
F(Du,u,x)
=
b
i
(x)∂
x
i
u
+
c(x)u
−
f(x).
i
=
0
(b
1
(x), b
2
(x))
=
(
1
,b)
,
b
Setting
x
0
=
0, for
example, we obtain the (hyperbolic) transport equation with constant speed
b
of
propagation
∂
t
u
t
,
x
1
=
x
,
b(x)
=
∈ R
+
, and
c
=
f(t,x)
.
For a
linear second order PDE
in
d
+
b∂
x
=
+
1 variables,
F
takes the form
d
d
F(D
2
u,Du,u,x)
=−
a
ij
(x)∂
x
i
x
j
u
+
b
i
(x)∂
x
i
u
+
c(x)u
−
f(x).
i,j
=
0
i
=
0
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