Optimal Control of Partial Differential Equations

28 CHAPTER 3. CONTROL OF SEMILINEAR ELLIPTIC EQUATIONS
3.2 Linear elliptic equations
Theorem 3.2.1 For every f ∈ L r (Ω), with r ≥ 2N
N+2
Az = f in Ω,
∂z
∂nA
admits a unique solution in H 1 (Ω), this solution belongs to W 2,r (Ω).
if N > 2 and r > 1 if N = 2, equation
= 0 on Γ, (3.2.2)
Proof. Since H1 (Ω) ⊂ L 2N
N−2 (Ω) (with a dense imbedding) if N > 2, and H1 (Ω) ⊂ Lp (Ω) for
any p < ∞ (with a dense imbedding) if N = 2, we have L 2N
N+2 (Ω) ⊂ (H1 (Ω)) ′ if N > 2, and
Lp′ (Ω) ⊂ (H1 (Ω)) ′ for all p < ∞ if N = 2. Thus Lr (Ω) ⊂ (H1 (Ω)) ′ . The existence of a unique
solution in H 1 (Ω) follows from the Lax-Migram theorem. The regularity result in W 2,r (Ω) is
proved in [30, Theorem 3.17].
For any exponent q, we denote by q ′ the conjugate exponent to q. When f ∈ (W 1,q′ (Ω)) ′ ,
q ≥ 2, we replace equation (3.2.2) by the variational equation
find z ∈ H 1 (Ω) such that a(z, φ) = 〈f, φ〉 (H 1 (Ω)) ′ ×H 1 (Ω) for all φ ∈ H 1 (Ω), (3.2.3)
where a(z, φ) =
Ω ΣN i,j=1aij(x)∂jz∂iφ dx.
Theorem 3.2.2 For every f ∈ (W 1,q′ (Ω)) ′ , with q ≥ 2, equation (3.2.3) admits a unique
solution in H1 (Ω), this solution belongs to W 1,q (Ω), and
z W 1,q (Ω) ≤ Cf (W 1,q ′ (Ω)) ′.
Proof. As previously we notice that the existence of a unique solution in H 1 (Ω) follows from
the Lax-Migram theorem. The regularity result in W 1,q (Ω) is proved in [30, Theorem 3.16].
With Theorem 3.2.2, we can study elliptic equations with nonhomogeneous boundary conditions.
Lemma 3.2.1 If g ∈ Ls (Γ) with s ≥ 2(N−1)
, the mapping
N
φ ↦−→ gφ
belongs to (W 1,q′
(Ω)) ′ for all s ≥ (N−1)q
N .
Proof. If φ ∈ W 1,q′
φ ↦→
1,q′
gφ belongs to (W (Ω)) ′ if s ≥
Γ
Γ
1
1−
(Ω), then φ|Γ belongs to W q ′ ,q ′
(Γ) ⊂ L (N−1)q′
N−q ′ (Γ). Thus the mapping
(N−1)q ′ ′
N−q ′
Theorem 3.2.3 For every g ∈ Ls (Γ), with s ≥ 2(N−1)
, equation
N
Az = 0 in Ω,
∂z
∂nA
= (N−1)q
. The proof is complete.
N
= g on Γ, (3.2.4)
admits a unique solution in H1 (Ω), this solution belongs to W 1,q (Ω) with q = Ns , and
N−1
z W 1,q (Ω) ≤ CgL s (Γ).