Chapter 6 Partial Differential Equations

6.3. CHARACTERISTICS AND HIGHER ORDER EQUATIONS 31
6.3 Characteristics and Higher Order Equations
6.3.1 Characteristics and Classification of 2nd Order Equations
L =
n∑ ∂ 2
a ij (x)
(6.3.1)
∂ xi ∂ xj
i,j=1
where a ij are real valued functions in Ω ⊂ R n and a ij = a ji . Fix a point x 0 ∈ Ω. The
characteristic polynomial is given by
We say that the operator L is:
σ x0 (L, ξ) =
n∑
a ij (x 0 )ξ i ξ j (6.3.2)
i,j=1
1. Elliptic at x 0 if the quadratic form (6.3.2) is non-singular and definite, i.e., can be
reduced by a real linear transformation to the form
n∑
ã i ξi 2 + l. o. t.
i=1
2. Hyperbolic at x 0 if the quadratic form (6.3.2) is non-singular and indefinite and can be
reduced by a real linear transformation to a sum of n squares, (n − 1) of the same sign,
i.e., to the form
n∑
ξ1 2 − ã i ξi 2 + l. o. t.
i=2
3. Ultra-Hyperbolic at x 0 if the quadratic form (6.3.2) is non-singular and indefinite and
can be reduced by a real linear transformation to a sum of n squares, (n ≥ 4) with
more than one terms of either sign.
4. Parabolic at x 0 if the quadratic form (6.3.2) is singular, i.e., can be reduced by a real
linear transformation to a sum of fewer than n squares, (not necessarily of the same
sign).
It can be shown that in the constant coefficient case a reduction to one of these forms is
always possible with a simple constant matrix transformation of coordinates.
The case of two independent variables and non-constant coefficients can also be analyzed.
a ∂2 u
∂x 2 +2b ∂2 u
∂x∂y + c∂2 u
∂y 2 + F (x, y, u, u x,u y ) = 0 (6.3.3)