Partial Differential Equations

44 2 First Order Equations
Let A ⊆ R m and A ′ ⊆ R m−1 be open sets, and let h: A ′ → R be a C 1 -function, whose graph lies in
A. We write
a = (a 1 , . . . , a m−1
} {{ }
=:a ′ , a m ) = (a ′ , a n ).
Now, let u: U × A → R be a family of solutions of F (Du, u, x) = 0. If the envelope v of the function
family
ũ(·; a ′ ): U → R,
x ↦→ u(x; a ′ , h(a ′ ))
exists, and if v is continuously differentiable, then it is called a general integral of F (Du, u, x) = 0 with
respect to h.
Example 2.1.7. We consider the Eikonal equation |Du| = 1 for n = 2. The family
u(x; a) = x 1 cos a 1 + x 2 sin a 1 + a 2
which is parameterized by R 2 is a complete integral. If we set h = 0, then we obtain a family
ũ(x; a 1 ) = x 1 cos a 1 + x 2 sin a 1 + 0
which is parameterized by R and which admits an envelope. To determine it, we calculate
which leads to a 1 = arctan x2
x 1
, and thus, to
v(x) = x 1 cos
D a1 ũ(x; a 1 ) = −x 1 sin a 1 + x 2 cos a 1 = 0
(
arctan x ) (
2
+ x 2 sin arctan x )
2
= ±|x|.
x 1 x 1
Example 2.1.8. We take up Example 2.1.4 with the Hamiltonian function H(p) = |p| 2 , and we again
choose h = 0. Then, we get
ũ(x, t; a) = x · a − t|a| 2
and D a ũ(x, t; a) = x − 2ta. For D a ũ(x, t; a) = 0, we then have a = x 2t
, and we obtain
as envelope for t > 0.
v(x, t) = x · x ∣ ∣∣
2t − t x
∣ 2 = |x|2
2t 4t
0.8
0.6
0.4
0.2
0
-1
-0.5
x
0
0.5
1
1
0.8
0.6
0.4
t
Fig. 2.4.