33.5 An Evolution Inclusion
In this section is a theorem on existence and uniqueness for the initial value
Suppose ϕ is a mapping from H to
which satisfy the following axioms.
Lemma 33.5.1 For x ∈ L2
,t → ϕ
Proof: This follows because ϕ is Borel measurable and so ϕ ∘ x is also measurable.
Now define the following function Φ, on the Hilbert space, L2
Lemma 33.5.2 Φ is convex, nonnegative, and lower semicontinuous on
Proof: Since ϕ is nonnegative and convex, it follows that Φ is also nonnegative and
convex. It remains to verify lower semicontinuity. Suppose, xn → x in L2
Is λ ≥ Φ
? Then is suffices to assume
λ < ∞.
Suppose not. Then λ <
subsequence, we can have λ
and we can take a further subsequence for
which convergence of
is pointwise a.e. Then
which is a contradiction. ■
and for x ∈ D
Then L is maximal monotone. To see this, consider the equation
It clearly has a solution so λL + I is onto. In fact, the solution is
Thus we have the following lemma.
Lemma 33.5.3 L is maximal monotone and if z ∈ L2
, then Jλz is given
The main theorem is the following.
Theorem 33.5.4 Let x0 ∈ D ≡ D
. Then L
Φ is maximal monotone so there
exists a unique solution to
for every f ∈ L2
. Thus there exists x ∈ L2
x0 ∈ D
Proof: This is from Theorem 33.4.2. Since x0 ∈ D, it follows that ϕ
Let z ∈ D
the effective domain of Φ.
dt < ∞,
so by convexity
The conditions of Theorem 33.4.2 are satisfied. This proves L + ∂Φ is maximal
monotone on L2
and consequently there exists a unique solution to the
differential inclusion of the theorem.
Then the main result is the following.
Theorem 33.5.5 Let f ∈ L2
and x0 ∈ D. Let ϕ be as described above, a lower
semicontinuous convex proper function defined on H. Then there exists a unique solution
x ∈ L2
This satisfies the pointwise condition
Proof: From Theorem 33.5.4, there exists a unique solution to
whenever v ∈ L2
. Then a simple argument based on fundamental theorem of
calculus implies that for a.e.
Then for given v,u one can act on xv
and integrate. This yields
It follows that a sufficiently high power of the mapping u → xu is a contraction map on
and so there exists a unique fixed point