Optimal Control of Partial Differential Equations

3.3. SEMILINEAR ELLIPTIC EQUATIONS 29
Proof. Obviously z ∈ H1 (Ω) is a solution to equation (3.2.4) if and only if
a(z, φ) = gφ for all φ ∈ H 1 (Ω).
Γ
The existence and uniqueness still follow from the Lax-Milgram theorem. The regularity result
is a direct consequence of Lemma 3.2.1 and Theorem 3.2.2.
3.3 Semilinear elliptic equations
The Minty-Browder Theorem, stated below, is a powerful tool to study nonlinear elliptic
equations.
Theorem 3.3.1 ([4]) Let E be a reflexive Banach space, and A be a nonlinear continuous
mapping from E into E ′ . Suppose that
and
〈A(z1) − A(z2), z1 − z2〉E ′ ,E > 0 for all z1, z2 ∈ E, with z1 = z2, (3.3.5)
〈A(z), z〉E
limzE→∞
′ ,E
zE
= ∞.
Then, for all ℓ ∈ E ′ , there exists a unique z ∈ E such that A(z) = ℓ.
We want to apply this theorem to the nonlinear equation
Az = f in Ω,
∂z
+ ψk(z) = g on Γ, (3.3.6)
∂nA
with f ∈ Lr (Ω), g ∈ Ls (Γ), and
⎧
⎨ ψ(k) + ψ
ψk(z) =
⎩
′ (k)(z − k) if z > k,
ψ(z)
ψ(−k) + ψ
if |z| ≤ k,
′ (−k)(z + k) if z < −k.
We explain below why we first replace ψ by the truncated function ψk (see remark after
Theorem 3.3.2). To apply Theorem 3.3.1, we set E = H1 (Ω), and we define A by
〈A(z), φ〉 (H1 (Ω)) ′ ,H1 (Ω) = a(z, φ) + ψk(z)φ,
and ℓ by
〈ℓ, φ〉 =
Ω
fφ +
Condition (3.3.5) is satisfied because a(z1 − z2, z1 − z2) ≥ mz1 − z22 H1 (Ω)
z2)(z1 − z2) ≥ 0 (indeed, the function ψk is increasing). Moreover
〈A(z), z〉 (H 1 (Ω)) ′ ,H 1 (Ω)
z H 1 (Ω)
Γ
gφ.
≥ mz H 1 (Ω) → ∞ as z H 1 (Ω) → ∞.
Γ
and
Γ ψk(z1 −