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3.3 Elliptic Equations

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We start with an elliptic equation, (B − AC < 0). We seek a change of independent
variables that will put Equation 3.1 in the form

u σσ + u ττ = G(σ, τ, u, u σ , u τ ) (3.16)

If we make the change of variables determined by
√ √
−B + i AC − B 2 −B − i AC − B 2
ξ x ψ x
= , = ,
ξ y A ψ y A
the equation will have the form

u ξψ = G(ξ, ψ, u, u ξ , u ψ ).

ξ and ψ are complex-valued. If we then make the change of variables

ξ + ψ ξ − ψ
σ = , τ =
2 2i
we will obtain the canonical form of Equation 3.16. Note that since ξ and ψ are complex
conjugates, σ and τ are real-valued.

Example 3.5 Consider
2
2
y u xx + x u yy = 0. (3.17)
For x 6= 0 and y 6= 0 this equation is elliptic. We find new variables that will put this
equation in the form u ξψ = G(·). From Example 3.2 we see that they are
p
2 2
dy y x x y 2 x 2 2 2
= −i = −i , y dy = −ix dx, = −i + const, ξ = y + ix
dx y 2 y 2 2
p
2 2
dy y x x y 2 x 2 2 2
= i = i , y dy = ix dx, = i + const, ψ = y − ix
dx y 2 y 2 2

The variables that will put Equation 3.17 in canonical form are
ξ + ψ ξ − ψ
2
σ = = y , τ = = x 2
2 2i
We calculate the derivatives of σ and τ.

σ x = 0 σ y = 2y
τ x = 2x τ y = 0

Then we calculate the derivatives of u.

u x = 2xu τ

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