Chapter 6 Partial Differential Equations

10 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS
Note that if ξ ≠ 0 is a vector in the x j -direction (i.e., ξ i = 0 for i ≠ j), then ξ ∈ Char x (L)
if and only if the coefficient of ∂j
k in L vanishes at x. Now, given any ξ ≠ 0, by a rotation
of coordinates we can arrange for ξ to lie in a coordinate direction. Thus the condition
ξ ∈ Char x (L) means that, in some sense, L fails to be “genuinely kth order” in the ξ
direction at x.
L is said to be elliptic at x if Char x (L) =∅ and elliptic on Ω if it elliptic at each x ∈ Ω.
Elliptic operators exert control on all derivatives of all order.
Example 6.2.3. The first three examples are in R 2 as discussed above.
1. L = ∂ 1 : Char x (L) ={ξ ≠0:ξ 1 =0}.
2. L = ∂ 1 ∂ 2 : Char x (L) ={ξ ≠0:ξ 1 =0 or ξ 2 =0}.
3. L = 1 2 (∂ 1 + i∂ 2 ): L is elliptic on R 2 .
4. L =
n∑
∂j 2 (Laplace Operator): L is elliptic on R n .
j=1
n∑
5. L = ∂ 1 − ∂j 2 (Heat Operator): Char x (L) ={ξ ≠0:ξ j =0, for j ≥ 2}.
j=2
6. L = ∂ 2 1 −
n∑
∂j
2
j=2
(Wave Operator): Char x (L) ={ξ ≠0:ξ 2 j = ∑ n
j=2 ξ2 j }.
Remark 6.2.4. In the notation introduced in Definition 6.2.1 we say that a surface S is
oriented if for each s ∈ S we have made a choice of a vector ν(x) which is orthogonal to S
and is a continuously varying function of x. Such a vector is called a normal vector to S at
x. OnS ∩ V = {x : ϕ(x) =0} we have
ν(x) =± ∇ϕ(x)
|∇ϕ(x)| .
Thus ν(x) is a C k−1 function on S. If S is the boundary of a domain Ω then we usually
choose the orientation so that ν points out of Ω.
At this point we can also define the normal derivative by
.
∂ ν u = ν ·∇u.