4 General Properties of Solutions of Classical Field Equations
19
u ∈ H
1
loc (R
s
) ⊕ L
2
loc (R
s
) ≡ X loc .
(4.7)
The space X loc is equipped with the natural topology generated by the family of
Hilbert seminorms
u
2
V =
V
((∇ϕ)
2
+ ϕ
2
)d
s x +
V
ψ
2 d
s x.
(4.8)
Thus, X loc is the Fréchet space defined as the inductive limit of the Hilbert spaces
H(V ) with Hilbert products (4.8).
As in the finite-dimensional case, in order to solve the Cauchy problem we need
some kind of Lipschitz condition
14 on the potential; in agreement with the local
structure discussed above, it is natural to choose the following local condition.
Local Lipschitz Condition
a) f (u) defines a continuous mapping of X loc into X loc
b) for any sphere Ω R , of radius R, and for any ρ > 0, there exists a constant
C(Ω R , ρ), such that
f (u 1 ) − f (u 2 ) Ω R ≤ C(Ω R , ρ)u 1 − u 2 Ω R ,
(4.9)
for all u 1 , u 2 ∈ X loc such that u i Ω R ≤ ρ, i = 1, 2 and
sup
0≤t≤R/2
C(Ω R−t , ρ) ≡ ¯
C(Ω R , ρ) < ∞.
The above local Lipschitz condition is satisfied by a large class of potentials U :
i) in s = 1 dimension, if U (ϕ) is an entire function;
ii) for s = 2, if
U (ϕ) =
∞
α∈N n
C α ϕ
α
,
(4.10)
α being a multi-index, ϕ
α
= ϕ
α 1
1 . . . ϕ
α n
n , with
α∈N n
|C α ||α|
|α|/2
|ϕ|
|α|
< ∞,
iii) for s = 3, if U is a twice differentiable real function such that
sup
ϕ
(1 + |ϕ|
2
)
−1
|U
(ϕ)| < ∞.
(4.11)
14 See e.g. V. Arnold, Ordinary Differential Equations, Springer 1992, Chap. 4; G. Sansone and R.
Conti, Non-linear Differential Equations, Pergamon Press 1964.