����������� ������ 53 (2001), 37–49

UDK 517.977

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SUFFICIENT CONDITIONS FOR ELLIPTIC PROBLEM OF OPTIMAL CONTROL IN R^{n} IN ORLICZ SOBOLEV SPACES

S. Lahrech and A. Addou

Abstract. This paper is concerned with the local minimization problem for a variety of non Frechet-differentiable Gâteaux functional J(f) ≡ ∫_{Ω} v(x, u, f) dx in the Orlicz-Sobolev space (W_{0}^{1}L_{M}^{∗} (Ω), ∥.∥_{M}), where u is the solution of the Dirichlet problem for a linear uniformly elliptic operator with nonhomogenous term f and ∥.∥_{M} is the Orlicz norm associated with an N-function M.

We use a recent extension of Frechet-Differentiability (approach of Taylor mappings see [2]), and we give various assumptions on v to guarantee a critical point is a strict local minimum.

Finally, we give an example of a control problem where classical Frechet differentiability cannot be used and their approach of Taylor mappings works.

1. Preliminaries

1.1. Some definitions and well-know facts from Orlicz space. We begin by listing briefly some definitions and well-known facts from Orlicz space theory (see [1]).

Let Ω be an open subset of R^{n}, with Lebesgue measure dx, and let M be an N-function (i.e. a real-valued continuous, convex, even function of t ∈ R satisfying M(t) > 0 for t > 0, \frac{M(t)}{t} → 0 as t → 0 and \frac{M(t)}{t} → +∞ as t → +∞).

The Orlicz class L_{M}(Ω) is defined as the set of (equivalence classes of) realvalued measurable functions u on Ω such that ∫_{Ω} M(u(x)) dx < +∞, and the Orlicz space L_{M}^{∗}(Ω) as the linear hull of L_{M}(Ω).

L_{M}^{∗}(Ω) is a Banach space with respect to the Luxembourg norm:

∥u∥_{(M)} = inf \{ k > 0 : ∫_{Ω} M(\frac{u(x)}{k}) dx ≤ 1\} .

L_{M}^{∗}(Ω) is a Banach space with respect to the Orlicz norm:

∥u∥_{M} = sup\{|∫_{Ω} u(x)v(x) dx| : ∫_{Ω} \overline{M}(v(x)) dx ≤ 1\},

where \overline{M} is the N-function conjugate to M.

AMS Subject Classification: 49 K 27

The Orlicz norm ∥.∥_{M} is equivalent to ∥.∥_{(M)}: ∥.∥_{(M)} ≤ ∥.∥_{M} ≤ 2∥.∥_{(M)}. Let W^{m}L_{M}^{∗}(Ω) be the Orlicz-Sobolev space of functions u such that u and its

distribution derivatives up to order m lie L_{M}^{∗}(Ω).

W^{m}L_{M}^{∗}(Ω) is a Banach space with respect to the norm:

∥u∥_{m,M} = ( ∑_{|α|≤m} ∥D^{α}u∥_{(M)}^{2})^{\frac{1}{2}} .

1.2. Orlicz-Sobolev Spaces. We define a further Orlicz-Sobolev space W_{0}^{m}L_{M}^{∗}(Ω) to be the closure of C_{0}^{∞}(Ω) in (W^{m}L_{M}^{∗}(Ω), σ(ΠL_{M}, ΠEM̄)).

1.3. Description of the optimization problem. Let A be an elliptic operator of second order:

Au ≡ ∑_{|l|≤1,|s|≤1}(−1)^{l}D^{l}(a_{ls}(x)D^{s}u),

where a_{ls}(x) ∈ D(\overline{Ω}). Suppose that Ω is sufficiently smooth and bounded domain in R^{n}.

Let us consider the problem :

Au = f,  (1.1)
u|_{∂Ω} = 0.  (1.2)

For this problem, let us state Agmon-Douglis-Niremberg’s theorem:

Theorem 1.1. (Agmon-Douglis-Niremberg) Let 1 < q < ∞; then we have that ∀f ∈ L^{q}(Ω), there exists a unique solution u ∈ W^{2,q}(Ω) ∩ W_{0}^{1,q}(Ω) of problem (1.1), (1.2). Moreover, ∀m ≥ 0 if f ∈ W^{m,q}(Ω), then u ∈ W^{m+2,q}(Ω) and ∥u∥_{W^{m+2,q}(Ω)} ≤ c∥f∥_{W^{m,q}(Ω)}.

Let M be an N-function such that |t|^{p} ≤ M(t) for t ≥ t_{0}, where p > n and t_{0} > 0. Let f ∈ F ⊂ W_{0}^{1}L_{M}^{∗}(Ω) be a control and let u the solution of problem (1.1), (1.2) in W_{0}^{1,p}(Ω) ∩ W^{2,p}(Ω) associated to f.

Let us consider J_{k}(f) = ∫_{Ω} v_{k}(x, u, f) dx + c_{k}∥f∥_{W^{1,2}(Ω)}^{2}, (k = 0, 1, 2,…, s_{1}) and J_{k}(f) = ∫_{Ω} v_{k}(x, u, f) dx, (k = s_{1} + 1, s_{1} + 2,…, s_{1} + s_{2}), where the sequence of functions v_{k} : Ω × R × R ⟶ R is measurable on Ω × R × R and has second derivative with respect to (u, f) on R × R for almost all x ∈ Ω.

We consider three problems of minimizing the functional J_{0}(f) :

i) J_{0}(f) → min,  (1.3)
ii) J_{0}(f) → min, J(f) = 0, where J = (J_{s_{1}+1},…, J_{s_{1}+s_{2}}),  (1.4)
iii) J_{0}(f) → min, J(f) = 0, J_{k}(f) ≤ 0, (k = 1, 2,…, s_{1}).  (1.5)

Elliptic problem of optimal control in R^{n} 39

We must choose a control f^{0} in order that the solution u^{0} of the problem (1.1), (1.2) with f = f^{0} satisfies the inequality of the type: J_{k}(f) ≤ 0, (1 ≤ k ≤ s_{1}) and the equality of the type: J_{k}(f) = 0, (s_{1} + 1 ≤ k ≤ s_{1} + s_{2}) and the functional J_{0}(f) takes a minimum value. This control f^{0} will be called optimal.

1.4. Taylor mappings and lower semi-Taylor mappings. Let M be an N-function, ∥.∥_{W^{1}L_{M}^{∗} (Ω)} the usual norm in W_{0}^{1}L_{M}^{∗}(Ω), F a subset of W_{0}^{1}L_{M}^{∗}(Ω), τ a topology in F, Y a normed space, and ∥.∥_{Y} a norm in Y. According to [2], a mapping r: F ⟶ Y (respectively, r: F ⟶ R) is said to be infinitesimally (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-small (respectively, infinitesimally lower (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})- semismall) of order p_{1} at f ∈ F if: ∀ε > 0, ∃ O_{f} ∈ τ, ∀h ∈ W_{0}^{1}L_{M}^{∗}(Ω) we have

f + h ∈ O_{f} ⇒ ∥r(f + h)∥_{Y} ≤ ε∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{p_{1}},

(respectively, ∀ ε > 0, ∃ O_{f} ∈ τ, ∀ h ∈ W_{0}^{1}L_{M}^{∗}(Ω) we have f + h ∈ O_{f} ⇒ ∥r(f + h)∥_{Y} ≥ −ε∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{p_{1}});

here and below, O_{f} is a neighborhood of f in (F, τ).

A mapping J: F → Y (respectively, J: F → R) is called a (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})- Taylor (respectively, lower (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})–semi-Taylor) mappings of order p_{1} at f ∈ F if there exist k linear symmetric (not necessarily continuous) mappings J^{(k)}(f): (W_{0}^{1}L_{M}^{∗}(Ω))^{k} → Y (respectively, J^{(k)}(f): (W_{0}^{1}L_{M}^{∗}(Ω))^{k} → R, k = 1,…, p_{1}, such that

J(f + h) − J(f) =

= J^{(1)}(f)h + 2^{−1}J^{(2)}(f)(h, h) + ⋯ + (p_{1})!^{−1}J^{(p_{1})}(f)(h,…, h) + r(f + h), where r: F ⟶ Y (respectively, r: F ⟶ R) is an infinitesimally (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})– small (respectively, infinitesimally lower (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-semismall) mapping of order p_{1} at f ∈ F.

We note that J^{(1)}(f),…, J^{(p_{1})}(f) are not in general single-valued. The set of tuples (J^{(1)}(f),…, J^{(p_{1})}(f)) is denoted by S_{n}(J, f).

Let us solve the problems (1.3), (1.4) and (1.5). For the problem (1.5) let us introduce the Lagrange functions:

L(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}(f) + ⟨y^{∗}, J(f)⟩,  (1.6)
L_{f}(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}^{(1)}(f) + ⟨y^{∗}, J^{(1)}(f)⟩,  (1.7)
L_{ff}(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}^{(2)}(f) + ⟨y^{∗}, J^{(2)}(f)⟩,  (1.8)

where λ_{0} ∈ R, y^{∗} ∈ (R^{s_{2}})^{∗}, λ ∈ (R^{s_{1}})^{∗}.

Also for the problem (1.4), let us introduce the Lagrange functions:

L(f, y^{∗}, λ_{0}) = λ_{0}J_{0}(f) + ⟨y^{∗}, J(f)⟩,  (1.9)
L_{f}(f, y^{∗}, λ_{0}) = λ_{0}J_{0}^{(1)}(f) + ⟨y^{∗}, J^{(1)}(f)⟩,  (1.10)
L_{ff}(f, y^{∗}, λ_{0}) = λ_{0}J_{0}^{(2)}(f) + ⟨y^{∗}, J^{(2)}(f)⟩,  (1.11)

where λ_{0} ∈ R, y^{∗} ∈ (R^{s_{2}})^{∗}.

Let us give the following theorem where the proof can be traced back to [1].

Theorem 1.2. Let M be an N-function, Ω a bounded domain in R^{n}. Suppose that u ∈ L_{M}^{∗}(Ω) and ∥u∥_{M} < 1, then ∫_{Ω} M(u(x)) dx < ∞.

Let us give also the following lemma where the proof can be traced back to [2].

Lemma 1.1. Let (Ω, Σ, μ) be a measure space with σ-finite measure, and let X be a complete linear metric space continuously imbedded in the metric space M(Ω) of equivalence classes of measurable almost everywhere finite functions x: Ω ⟶ R, with the metrizable topology τ(meas) of convergence in measure on each set of Σ finite measure.

Suppose that X contains with each element x(s) the function |x(s)|, the metric in X is translation-invariant, and ρ(x, 0) = ρ(|x|, 0) for each x ∈ X. Then for each sequence x_{n} → 0 in X there exist a subsequence x_{n_{k}} and an element z ∈ X such that: |x_{n_{k}}(s)| ≤ z(s), k=1,2, … in the sense of the natural order on classes of functions.

Using the same argument as in Lemma 1.1 and the result of Theorem 1.2, we obtain the following lemma.

Lemma 1.2. Let M be an N-function, Ω a bounded domain in R^{n}. Then for each sequence u_{n} → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}) there exist a subsequence u_{n_{k}} and an element z ∈ L_{M}^{∗}(Ω) such that |u_{n_{k}}(s)| ≤ z(s), k = 1, 2,…, in the sense of the natural order on classes of functions. Moreover, 2z ∈ L_{M}(Ω) i.e. ∫_{Ω} M(2z(x)) dx < +∞.

Proof. Suppose that u_{n} → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}). Then, there exists a subsequence u_{n_{k}} such that ∥u_{n_{k}}∥_{M} ≤ \frac{1}{2^{k}}. Let us put S_{n}(x) = ∑_{k=0}^{n} |u_{n_{k}}(x)|. We show that the sequence S_{n}(x) is Cauchy in (L_{M}^{∗}(Ω), ∥.∥_{M}). Let m > n. Then

∥S_{n}(x) − S_{m}(x)∥_{M} = ∥ ∑_{k=n+1}^{m} |u_{n_{k}}(x)|∥_{M} ≤ 2 ∑_{k=n+1}^{m} ∥u_{n_{k}}(x)∥_{M} ≤ 2 ∑_{k=n+1}^{m} \frac{1}{2^{k}} .

Since (L_{M}^{∗}(Ω), ∥.∥_{M}) is complete, S_{0}(x) = ∑_{k=1}^{+∞} |u_{n_{k}}(x)| ∈ L_{M}^{∗}(Ω). Consequently, there exists k_{0} ∈ N such that ∥ ∑_{k=k_{0}}^{+∞} |u_{n_{k}}(x)|∥_{M} < \frac{1}{2}.

It can be assumed that S_{n}(x) → S_{0}(x) almost everywhere in Ω. Let us put Z(x) = ∑_{k=k_{0}}^{+∞} |u_{n_{k}}(x)|. Obviously, ∀k ≥ k_{0} |u_{n_{k}}(x)| ≤ Z(x) almost everywhere in Ω. Further, ∥Z(x)∥_{M} < \frac{1}{2}. By theorem (1.2), it follows that ∫_{Ω} M(2z(x)) dx < +∞. Thus we achieve the proof.

Lemma 1.3. Suppose that Ω is sufficiently smooth and bounded domain in R^{n}. Let M be an N-function such that |t|^{p} ≤ M(t) for t ≥ t_{0}, where p > n and t_{0} > 0. Then, ∃ c > 0 ∀f ∈ W_{0}^{1}L_{M}^{∗}(Ω) ∥R(f)∥_{C(Ω̄)} ≤ c∥f∥_{W^{1}L_{M}^{∗} (Ω)} and ∥f∥_{C(Ω̄)} ≤ c∥f∥_{W^{1}L_{M}^{∗} (Ω)}, where R(f) is the solution of problem (1.1), (1.2) in W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω).

Proof. Let f ∈ W_{0}^{1}L_{M}^{∗}(Ω). Since |t|^{p} ≤ M(t), then f ∈ L^{p}(Ω). Thus, there exists a unique solution u ∈ W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω) of problem (1.1), (1.2). Moreover, ∥u∥_{W^{1,p}(Ω)} ≤ c∥f∥_{L^{p}(Ω)}, where c > 0.

On the other hand, W_{0}^{1,p}(Ω) ↪ C(Ω̄). Therefore, u ∈ C(Ω̄). Let us put R(f) = u. So, since W_{0}^{1,p}(Ω) ↪ C(Ω̄), then

∥R(f)∥_{C(Ω̄)} ≤ c_{1}∥R(f)∥_{W^{1,p}(Ω)} ≤ c_{2}∥f∥L^{p}(Ω) ≤ c_{3}∥f∥_{L_{M}^{∗} (Ω)} ≤ c_{4}∥f∥_{W^{1}L_{M}^{∗} (Ω)},

where c_{1}, c_{2}, c_{3}, c_{4} > 0.

On the other hand, we have W_{0}^{1}L_{M}^{∗}(Ω) ↪ W_{0}^{1,p}(Ω) ↪ C(Ω̄). Thus, there exists c_{5} > 0 such that ∥f∥_{C(Ω̄)} ≤ c_{5}∥f∥_{W^{1}L_{M}^{∗} (Ω)}.

2. Sufficient conditions of local minimum for Gâteaux functional of second order Dirichlet problem

Suppose that Ω is sufficiently smooth and bounded domain in R^{n}. Let F be a subset of W_{0}^{1}L_{M}^{∗}(Ω), M an N-function such that |t|^{p} ≤ M(t) for t ≥ t_{0}, where p > n and t_{0} > 0. Let G be the functional defined on W_{0}^{1}L_{M}^{∗}(Ω) by: G(f) = ∫_{Ω} v(x, u(x), f(x)) dx, where u(x) is the solution of problem (1.1), (1.2) in W_{0}^{1,p}(Ω)∩W^{2,p}(Ω) and the function v: Ω×R×R → R is measurable on Ω×R×R and has second derivative with respect to (u, f) on R × R for almost all x ∈ Ω. Let τ_{M} be the topology generated by the Orlicz norm ∥.∥_{M}. Henceforth in this paragraph a = const.

Theorem 2.1. Suppose that the following condition is added to the conditions of paragraph (1) and (2): v, v_{uf}^{(2)}, v_{fu}^{(2)} are continuous in Ω × R × R. Let us suppose also that

|v(x, u, f)| + |v_{u}^{(1)}(x, u, f)| + |v_{f}^{(1)}(x, u, f)| ≤ a(M(u) + M(f)) + b_{5}(x),
|v_{uu}^{(2)}(x, u, f)| + 2|v_{uf}^{(2)}(x, u, f)| + |v_{ff}^{(2)}(x, u, f)| ≤ a(M(u) + M(f)) + b_{6}(x),

where b_{5} ∈ L^{1}(Ω), b_{6} ∈ L^{1}(Ω). Then, G is a (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-Taylor mapping of first and second order at each point f ∈ F. Moreover, G^{(2)}(f) ∈ B((W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)}), R), G^{(1)}(f) ∈ L((W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)}), R).

Proof. Let us prove first that the functional G is finite. We have

|G(f)| = |∫_{Ω} v(x, u, f) dx| ≤ ∫_{Ω} |v(x, u, f)| dx
≤ a(∫_{Ω} M(u(x)) dx + ∫_{Ω} M(f(x)) dx) + ∫_{Ω} b_{5}(x) dx < ∞.

Indeed, we have f ∈ F ⊂ W_{0}^{1}L_{M}^{∗}(Ω) ↪ C(Ω̄). Consequently, f ∈ C(Ω̄). On the other hand, u ∈ W_{0}^{1,p}(Ω) ↪ C(Ω̄). Therefore, ∫_{Ω} M(u(x)) dx < ∞ and ∫_{Ω} M(f(x)) dx < ∞. Thus, the functional G is finite.

Let R: W_{0}^{1}L_{M}^{∗}(Ω) ⟶ W_{0}^{1}L_{M}^{∗}(Ω), h ⟼ (R(h))(x), where (R(h))(x) is the solution of problem

Au = h,  (2.1)
u|_{∂Ω} = 0.  (2.2)

Such a solution exists ∀h ∈ W_{0}^{1}L_{M}^{∗}(Ω).

Let G^{(1)}(f) be defined by:

G^{(1)}(f)h = lim_{λ→0} λ^{−1}(G(f + λh) − G(f))

= lim_{λ→0} λ^{−1} ∫_{Ω}[v(x, u + λR(h), f + λh) − v(x, u, f)] dx

= lim_{λ→0} λ^{−1} ∫_{Ω}[v(x, u + λR(h), f + λh)−

− v(x, u, f + λh) + v(x, u, f + λh) − v(x, u, f)] dx

= lim_{λ→0} ∫_{Ω}[∫_{0}^{1} v_{u}^{(1)}(x, u + θλR(h), f + λh)R(h) dθ

+ ∫_{0}^{1} v_{f}^{(1)}(x, u, f + ρλh)h dρ]dx

= lim_{λ→0} ∫_{Ω}[∫_{0}^{1}[v_{u}^{(1)}(x, u + θλR(h), f + λh) − v_{u}^{(1)}(x, u, f)]R(h) dθ

+ ∫_{0}^{1} v_{u}^{(1)}(x, u, f)R(h) dθ + ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + ρλh) − v_{f}^{(1)}(x, u, f)]h dρ
+ h ∫_{0}^{1} v_{f}^{(1)}(x, u, f)dρ] dx

= ∫_{Ω} v_{u}^{(1)}(x, u, f)R(h) dx + ∫_{Ω} hv_{f}^{(1)}(x, u, f) dx.

Then

G^{(1)}(f) = ∫_{Ω} v_{u}^{(1)}(x, u, f)R(h) dx + ∫_{Ω} v_{f}^{(1)}(x, u, f)h dx.

Let G^{(2)}(f) be defined by:

G^{(2)}(f)(h_{1}, h_{2}) = lim_{λ→0} λ^{−1}[G^{(1)}(f + λh_{2}) − G^{(1)}(f)]h_{1}

= lim_{λ→0} λ^{−1}[∫_{Ω}[v_{u}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{u}^{(1)}(x, u, f)]R(h_{1}) dx

+ ∫_{Ω}[v_{f}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{f}^{(1)}(x, u, f)]h_{1} dx]

= lim_{λ→0} λ^{−1}[∫_{Ω}[v_{u}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{u}^{(1)}(x, u, f + λh_{2})

+ v_{u}^{(1)}(x, u, f + λh_{2}) − v_{u}^{(1)}(x, u, f)]R(h_{1}) dx

+ ∫_{Ω}[v_{f}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{f}^{(1)}(x, u, f + λh_{2})

+ v_{f}^{(1)}(x, u, f + λh_{2}) − v_{f}^{(1)}(x, u, f)]h_{1}dx]

= lim_{λ→0} λ^{−1}[∫_{Ω}[∫_{0}^{1} v_{uu}^{(2)}(x, u + θλR(h_{2}), f + λh_{2})λR(h_{2}) dθ

+ ∫_{0}^{1} v_{fu}^{(2)}(x, u, f + ρλh_{2})λh_{2} dρ]R(h_{1}) dx

+ ∫_{Ω}[∫_{0}^{1} v_{uf}^{(2)}(x, u + θλR(h_{2}), f + λh_{2})λR(h_{2}) dθ

+ ∫_{0}^{1} v_{ff}^{(2)}(x, u, f + ρλh_{2})λh_{2} dρ]h_{1} dx]

= ∫_{Ω} v_{uu}^{(2)}(x, u, f)R(h_{1})R(h_{2}) dx + ∫_{Ω} v_{uf}^{(2)}(x, u, f)R(h_{1})h_{2} dx

+ ∫_{Ω} v_{fu}^{(2)}(x, u, f)h_{1}R(h_{2}) dx + ∫_{Ω} v_{ff}^{(2)}(x, u, f)h_{1}h_{2} dx.

The linearity and bilinearity of G^{(1)}(f) and G^{(2)}(f) are obvious.

Let us prove now that they are bounded.

|G^{(1)}(f)h| ≤ ∫_{Ω} |v_{u}^{(1)}(x, u, f)||R(h)| dx + ∫_{Ω} |v_{f}^{(1)}(x, u, f)||h| dx

≤ ∫_{Ω}[a(M(u) + M(f)) + |b_{5}(x)|][|R(h)| + |h)|] dx

≤ ∫_{Ω}[a(M(u) + M(f)) + |b_{5}(x)|][max_{x∈Ω̄} |[R(h)](x)| + max_{x∈Ω̄} |h(x)|] dx

= ∫_{Ω}[a(M(u) + M(f)) + |b_{5}(x)|][∥R(h)∥_{C(Ω̄)} + ∥h∥_{C(Ω̄)}] dx

≤ c_{2}∥h∥_{W_{1}L_{M}^{∗} (Ω)},

where c_{2} > 0. For the last inequality, see Lemma 1.3. Consequently, G^{(1)}(f) ∈ L((W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)}), R).

Let us prove now that G^{(2)}(f) is also bounded. We have

|G^{(2)}(f)(h_{1}, h_{2})| ≤ ∫_{Ω}[a(M(u) + M(f)) + |b_{6}(x)|]×

× [|R(h_{1})||R(h_{2})| + |R(h_{1})||h_{2}| + |h_{1}||R(h_{2})| + |h_{1}||h_{2}|] dx ≤ [a(M(u) + M(f)) + ∥b_{6}(x)∥_{L_{1}(Ω)}]×

× [max_{x∈Ω̄} |[R(h_{1})](x)| max_{x∈Ω̄} |[R(h_{2})](x)| + max_{x∈Ω̄} |[R(h_{1})](x)| max_{x∈Ω̄} |h_{2}(x)|

+ max_{x∈Ω̄} |h_{1}(x)| max_{x∈Ω̄} |[R(h_{2})](x)| + max_{x∈Ω̄} |h_{1}(x)| max_{x∈Ω̄} |h_{2}(x)|]

≤ c_{3}[∥R(h_{1})∥_{C(Ω̄)}∥R(h_{2})∥_{C(Ω̄)} + ∥R(h_{1})∥_{C(Ω̄)}∥h_{2}∥_{C(Ω̄)}
+ ∥h_{1}∥_{C(Ω̄)}∥R(h_{2})∥_{C(Ω̄)} + ∥h_{1}∥_{C(Ω̄)}∥h_{2}∥_{C(Ω̄)}]
≤ c_{4}∥h_{1}∥_{W^{1}L_{M}^{∗} (Ω)}∥h_{2}∥_{W^{1}L_{M}^{∗} (Ω)}.

For the last inequality see Lemma 1.3. Thus,

G^{(2)}(f) ∈ B((W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)}), R).

Let us prove now that G is a (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-mapping. Let f ∈ F. We show that the mapping

r(h) ≡ G(f + h) − G(f) − G^{(1)}(f)h − 2^{−1}G^{(2)}(f)(h, h)

is (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-of second order at zero.

Assume that this is not so. Then there exist a sequence h̃_{m} ∈ F and a number ε > 0 such that h̃_{m} → 0 in L_{M}^{∗}(Ω), but

|r(h̃_{m})| ≥ ε∥h̃_{m}∥_{W^{1}L_{M}^{∗} (Ω)}^{2}.

On the other hand, using the fact that W_{0}^{1,p}(Ω) ↪ C(Ω̄) and the regularity of solution of the problem (1.1), (1.2), we obtain R(h̃_{m}) → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}). Using Lemma (1.2), we deduce that ∃z̃ ∈ L_{M}^{∗}(Ω) ∀m |(R(h̃_{m}))(x)| ≤ z̃(x), where 2z̃ ∈ L_{M}(Ω).

Analogously, for h̃_{m} ∈ W_{0}^{1}L_{M}^{∗}(Ω), we obtain |h̃_{m}(x)| ≤ z̃_{1}(x), where 2z̃_{1} ∈ L_{M}(Ω). We have

r(h) = ∫_{Ω}[v(x, u + R(h), f + h) − v(x, u, f) − v_{u}^{(1)}(x, u, f)R(h) − v_{f}^{(1)}(x, u, f)h
− 2^{−1}[v_{uu}^{(2)}(x, u, f)R^{2}(h) + 2v_{uf}^{(2)}(x, u, f)R(h)h + v_{ff}^{(2)}(x, u, f)h^{2}]] dx
= ∫_{Ω} ∫_{0}^{1}[v_{u}^{(1)}(x, u + θR(h), f + h)R(h) − v_{u}^{(1)}(x, u, f)R(h)

− 2^{−1}v_{uu}^{(2)}(x, u, f)R^{2}(h)] dθ dx + ∫_{Ω} ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + λh)h − v_{f}^{(1)}(x, u, f)h

− 2^{−1}v_{ff}^{(2)}(x, u, f)h^{2}] dλ dx − ∫_{Ω} ∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)h dλ dx
+ ∫_{Ω} ∫_{0}^{1} ∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h) dλ dθdx

= ∫_{Ω} ∫_{0}^{1}[v_{u}^{(1)}(x, u + θR(h), f) − v_{u}^{(1)}(x, u, f) − θv_{uu}^{(2)}(x, u, f)R(h)]R(h) dθdx

+ ∫_{Ω} ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + λh) − v_{f}^{(1)}(x, u, f) − λv_{ff}^{(2)}(x, u, f)h]h dλ dx
− ∫_{Ω}∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)h dλ dx

+ ∫_{Ω}∫_{0}^{1}∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h) dλ dθ dx.

Let A_{m}, B_{m} be two functions defined by:

A_{m}(x, θ) = \{ \begin{matrix}\frac{v_{u}^{(1)}(x,u+θR(h̃_{m}),f)−v_{u}^{(1)}(x,u,f)}{R(h̃_{m})} − θv_{uu}^{(2)}(x, u, f), & if R(h̃_{m}) ≠ 0, \\ 0, & if R(h̃_{m}) = 0, \end{matrix}

B_{m}(x, λ) = \{ \begin{matrix}\frac{v_{f}^{(1)}(x,u,f+λh̃_{m})−v_{f}^{(1)}(x,u,f)}{h̃_{m}} − λv_{ff}^{(2)}(x, u, f), & if h̃_{m} ≠ 0, \\ 0, & if h̃_{m} = 0. \end{matrix}

Let F_{m} be defined by: F_{m}(x, θ, λ) = v_{uf}^{(2)}(x, u(x)+θR(h̃_{m}), f+λh̃_{m})−v_{uf}^{(2)}(x, u(x), f). Then

|r(h̃_{m})| = |∫_{Ω} ∫_{0}^{1} A_{m}(x, θ)R^{2}(h̃_{m}) dθ dx + ∫_{Ω} ∫_{0}^{1} B_{m}(x, λ)h̃_{m}^{2} dλ dx
+ ∫_{Ω} ∫_{0}^{1} ∫_{0}^{1} F_{m}(x, θ, λ)R(h̃_{m})h̃_{m} dλ dθ dx|.

Thus

|r(h̃_{m})| ≤ ∫_{0}^{1} ∫_{Ω} |A_{m}(x, θ)| dx dθ max_{x∈Ω̄} |[R(h̃_{m})](x)|^{2}

+ ∫_{0}^{1} ∫_{Ω} |B_{m}(x, λ)| dx dλ max_{x∈Ω̄} |h̃_{m}|^{2}

+ ∫_{0}^{1} ∫_{0}^{1} ∫_{Ω} |F_{m}(x, θ, λ)| dx dλ dθ max_{x∈Ω̄} |[R(h̃_{m})](x)| max_{x∈Ω̄} |h̃_{m}|

≤ c_{5}[∫_{0}^{1} ∫_{Ω} |A_{m}(x, θ)| dx dθ + ∫_{0}^{1} ∫_{Ω} |B_{m}(x, λ)| dx dλ
+ ∫_{0}^{1} ∫_{0}^{1} ∫_{Ω} |F_{m}(x, θ, λ)| dx dλ dθ]∥h̃_{m})∥_{W^{1}L_{M}^{∗} (Ω)}^{2}.  (2.3)

On the other hand, ∃k_{m}(x) such that 0 ≤ k_{m}(x) ≤ 1 and

|A_{m}(x, θ)| ≤ |v_{uu}^{(2)}(x, u(x) + k_{m}(x)θ[R(h̃_{m})](x), f)| + |v_{uu}^{(2)}(x, u(x), f)|.

So, we obtain

|A_{m}(x, θ)| ≤ a[M(u(x) + k_{m}(x)θ[R(h̃_{m})](x)) + M(f)] + |b_{6}(x)|
+ a[M(u) + M(f)] + |b_{6}(x)|

≤ a[\frac{1}{2}M(2z̃(x))| + 2M(f) + M(u(x)) + \frac{1}{2}M(2u(x)))]

+ 2|b_{6}(x)| ∈ L^{1}(Ω).

Analogously, ∃S_{m}(x) such that 0 ≤ S_{m}(x) ≤ 1 and

|B_{m}(x, λ)| ≤ |v_{ff}^{(2)}(x, u(x), λS_{m}(x)h̃_{m}(x) + f)| + |v_{ff}^{(2)}(x, u(x), f)|.

So, we obtain

|B_{m}(x, λ)| ≤ a[M(u(x)) + M(λS_{m}(x)h̃_{m}(x) + f)] + |b_{6}(x)|
+ a[M(u(x)) + M(f(x))] + |b_{6}(x)|

≤ a[2M(u) + \frac{1}{2}M(2z̃_{1}(x)) + \frac{1}{2}M(2f(x)) + M(f)] + 2|b_{6}(x)| ∈ L^{1}(Ω).

Analogously for F_{m}, we obtain

|F_{m}(x, θ, λ)| ≤ |v_{uf}^{(2)}(x, u(x), f)| + |v_{uf}^{(2)}(x, u(x) + θR(h̃_{m}), f + λh̃_{m})|

≤ a[\frac{1}{2}M(2z̃(x)) + \frac{1}{2}M(2z̃_{1}(x))\frac{1}{2}M(2u) + \frac{1}{2}M(2f)] + 2|b_{6}(x)|

+ a[M(u) + M(f)] ∈ L^{1}(Ω).

Let us remark that A_{m}(x, θ) → 0, B_{m}(x, λ) → 0, F_{m}(x, θ, λ) → 0 almost everywhere. Thus

∫_{0}^{1} ∫_{Ω} |A_{m}(x, θ)| dx dθ → 0 as m → +∞,
∫_{0}^{1} ∫_{Ω} |B_{m}(x, λ)| dx dλ → 0 as m → +∞,
∫_{0}^{1} ∫_{0}^{1} ∫_{Ω} |F_{m}(x, θ, λ)| dx dλ dθ → 0 as m → +∞;

but this contradicts (2.3).

Now let us prove that the functional G is a (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-Taylor mapping of first order at each point f ∈ F. We must estimate

r(h) ≡ G(f + h) − G(f) − G^{(1)}(f)h.

Assume that h̃_{m} → 0 in L_{M}^{∗}(Ω), and |r(h̃_{m})| ≥ ε∥h̃_{m}∥_{W^{1}L_{M}^{∗} (Ω)}. Thus R(h̃_{m}) → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}) as m → +∞. Using Lemma (1.2), we deduce that there exists z̃ ∈ L_{M}^{∗}(Ω) and there exists z̃_{1} ∈ L_{M}^{∗}(Ω) such that ∀m ∈ N |[R(h̃_{m})](x)| ≤ z̃(x) and |h̃_{m}(x)| ≤ z̃_{1}(x) almost everywhere in Ω. Moreover, 2z̃ ∈ L_{M}(Ω) and 2z̃_{1} ∈ L_{M}(Ω).

On the other hand, we have

|r(h̃_{m})| = |∫_{Ω}[v(x, u(x) + [R(h̃_{m})](x), f(x) + h̃_{m}(x)) − v(x, u(x), f(x))
− v_{u}^{(1)}(x, u(x), f(x))R(h̃_{m}) − v_{f}^{(1)}(x, u(x), f(x))h̃_{m}] dx|
≤ |∫_{Ω}[v(x, u(x) + R(h̃_{m}), f(x) + h̃_{m}) − v(x, u(x), f(x) + h̃_{m})
− v_{u}^{(1)}(x, u(x), f(x))R(h̃_{m}) + v(x, u(x), f(x) + h̃_{m})

− v(x, u(x), f(x)) − v_{f}^{(1)}(x, u(x), f(x))h̃_{m}] dx|
= |∫_{Ω} A_{m}(x)R(h̃_{m}) dx| + |∫_{Ω} B_{m}(x)h̃_{m}dx|,

where

A_{m}(x) = \{ \begin{matrix}\frac{v(x,u+R(h̃_{m}),f+h̃_{m})−v(x,u,f+h̃_{m})}{R(h̃_{m})} − v_{u}^{(1)}(x, u, f), & if R(h̃_{m}) ≠ 0, \\ 0, & if R(h̃_{m}) = 0, \end{matrix}
B_{m}(x) = \{ \begin{matrix}\frac{v(x,u,f+h̃_{m})−v(x,u,f)}{h̃_{m}} − v_{f}^{(1)}(x, u, f), & if h̃_{m} ≠ 0, \\ 0, & if h̃_{m} = 0. \end{matrix}

Consequently,

|r(h̃_{m})| ≤ ∫_{Ω} |A_{m}(x)| dx max_{x∈Ω̄}|[R(h̃_{m})](x)| + ∫_{Ω} |B_{m}(x)|dx max_{x∈Ω̄}|h̃_{m}(x)|
≤ c_{6}[∫_{Ω} |A_{m}(x)| dx + ∫_{Ω} |B_{m}(x)| dx] ∥h̃_{m}∥_{W^{1}L_{M} (Ω)}.  (2.4)

Let us remark that A_{m}(x) → 0, B_{m}(x) → 0 almost everywhere in Ω.

Using the mean value theorem, we obtain

|A_{m}(x)| ≤ a[M(u + R(h̃_{m})) + M(f(x) + h̃_{m}(x))] + |b̂_{3}(x)|
+ a[M(u(x)) + M(f(x))] + |b̂_{3}(x)|

≤ a[\frac{1}{2}M(2Z̃(x)) + \frac{1}{2}M(2Z̃_{1}(x))] + 2|b̂_{3}(x)|

+ a[\frac{1}{2}M(2u(x)) + \frac{1}{2}M(2f(x))] + a[M(u(x)) + M(f(x))] ∈ L^{1}(Ω).

Using the same reasons, we obtain

|B_{m}(x)| ≤ a[M(u(x)) + M(f(x) + h̃_{m}(x))] + |b̂_{3}(x)|
+ a[M(u(x)) + M(f(x))] + |b̂_{3}(x)|

≤ a[2M(u(x)) + \frac{1}{2}M(2z̃_{1}(x)) + \frac{1}{2}M(2f(x))]

+ aM(f(x)) + 2|b̂_{3}(x)| ∈ L^{1}(Ω).

By the Lebesgue dominated convergence theorem, we conclude that

∫_{Ω} |A_{m}(x)| dx → 0 as m → +∞

and

∫_{Ω} |B_{m}(x)| dx → 0 as m → +∞;

but this contradicts (2.4).

Now let us give the sufficient conditions of optimality for the problem (1.3), (1.4) and (1.5).

Theorem 2.2 Suppose that, in problem (1.4), v_{k} satisfies the conditions of Theorem 2.1, then the functionals

J_{k}(f) ≡ ∫_{Ω} v_{k}(x, u, f) dx, (k = s_{1} + 1,…, s_{1} + s_{2})

are (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-Taylor mappings of first and second order at each point f ∈ F and J_{k}(f) = ∫_{Ω} v_{k}(x, u, f) dx + c_{k}∥f∥_{W^{1,2}(Ω)}^{2}, (k = 0,…, s_{1}) are lower (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-semi-Taylor mappings of first and second order at each point

f ∈ F. Consequently, ∃J_{k}^{(1)}(f) and ∃J_{k}^{(2)}(f), (k = 0,…, s_{1} + s_{2}).

Let us suppose also that, J(f̂) = 0, J^{(1)}(f̂) is an open mapping of

(W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)})

onto R^{s_{2}}, ∃ y^{∗} ∈ (R^{s_{2}})^{∗}, ∃α > 0: L_{f}(f̂, y^{∗}, 1) = 0, and ∀h ∈ kerJ^{(1)}(f̂) L_{ff}(f̂, y^{∗}, 1)(h, h) ≥ 2α∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{2}, where L_{f}(f̂, y^{∗}, 1) and L_{ff}(f̂, y^{∗}, 1) are given

by formulas (1.10), (1.11). Then f̂ is a strict τ_{M}-local minimum point.

Proof. All conditions of Theorem 1.5 in [2] are satisfied. Thus f̂ is a strict τ_{M}-minimum point.

Theorem 2.3. Suppose that, in problem (1.5), v_{k} satisfies the conditions of Theorem 2.1, then the functionals

J_{k}(f) ≡ ∫_{Ω} v_{k}(x, u, f) dx, (k = s_{1} + 1,…, s_{1} + s_{2})

are (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-Taylor mappings of first and second order at each point f ∈ F and J_{k}(f) = ∫_{Ω} v_{k}(x, u, f) dx + c_{k}∥f∥_{W^{1,2}(Ω)}^{2}, (k = 0,…, s_{1}) are lower (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-semi-Taylor mappings of first and second order at each point

f ∈ F. Consequently, ∃J_{k}^{(1)}(f) and ∃J_{k}^{(2)}(f), (k = 0,…, s_{1} + s_{2}).

Let us suppose also that, f̂ ∈ F, J(f̂) = 0, J_{k}(f̂) = 0, (k = 0,…, s_{1}). Let us put L = {h ∈ W_{0}^{1}L_{M}^{∗}(Ω)/J_{k}^{(1)}(f̂)h = 0, k = 1,…, s_{1}, J^{(1)}(f̂)h = 0}. Suppose that J^{(1)}(f̂) is an open map from (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) onto R^{s_{2}}, ∃λ̂ ∈ (R^{s_{1}})^{∗}, ∃y^{∗} ∈ (R^{s_{2}})^{∗}, ∃γ ≥ 0, ∃λ̂_{k} > 0, (k = 1,…, s_{1}): L_{f}(f̂, y^{∗}, λ̂, 1) = 0 and ∀h ∈ L L_{ff}(f̂, y^{∗}, λ̂, 1)(h, h) ≥ 2γ∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{2}, where L_{f}(f̂, y^{∗}, λ̂, 1) and L_{ff}(f̂, y^{∗}, λ̂, 1)

are defined by formulas (1.7) and (1.8). Then f̂ is a strict τ_{M}-local minimum point.

Proof. All conditions of Theorem 1.6 in [2] are satisfied. Thus f̂ is a strict τ_{M}-local minimum point.

Theorem 2.4. Suppose that, in problem (1.3), v_{k} satisfies the conditions of Theorem 2.1, then the functionals

J_{k}(f) ≡ ∫_{Ω} v_{k}(x, u, f) dx, (k = s_{1} + 1,…, s_{1} + s_{2})

are (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-Taylor mappings of first and second order at each point f ∈ F and J_{k}(f) = ∫_{Ω} v_{k}(x, u, f) dx + c_{k}∥f∥_{W^{1,2}(Ω)}^{2}, (k = 0,…, s_{1}) are lower (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)})-semi-Taylor mappings of first and second order at each point

f ∈ F. Consequently, ∃J_{k}^{(1)}(f) and ∃J_{k}^{(2)}(f), (k = 0,…, s_{1} + s_{2}).

Let us suppose also that, J_{0}^{(1)}(f̂) = 0 and ∃α > 0 ∀h ∈ W_{0}^{1}L_{M}^{∗}(Ω): J_{0}^{(2)}(f̂)(h, h) ≥ 2α∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{2}. Then f̂ is a strict τ_{M}-local minimum point.

Proof. All conditions of Theorem 1.4 in [2] are satisfied. Thus f̂ is a strict τ_{M}-local minimum point.

Remark. Let us remark that, in Theorem 2.1, the increasing conditions satisfied by v are not sufficient to certify the Frechet-differentiability of functional G : (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{M}) → R.

Indeed, let us define v: Ω × R × R → R by: v(x, u, f) = [ch|u| − |u| − 1] + [ch|f| − |f| − 1], and put b_{0}(x) ≡ 0. Let us suppose that a = 1. Let us put M(t) = e^{|t|} − |t| − 1, and let d_{m} → +∞. Let f̃ ∈ W_{0}^{1}L_{M}^{∗}(Ω). By the countable additivity of Lebesgue measure,

∃c > 0 ∃Ω′ ⊂ Ω, μ(Ω′) > 0, ρ(Ω′, ∂Ω) > 0,

and ∀x ∈ Ω′ |f̃(x)| ≤ c. Let Ω_{m} ⊂ Ω′ such that μ(Ω_{m}) = (e^{d_{m}} − d_{m} − 1)^{−1}. Put D ≡ max\{|v(x, u, f)| : |u| ≤ c, |f| ≤ c, x ∈ \overline{Ω}\} < ∞.

Let h̃_{m} be defined by:

h̃_{m}(x) = \{ \begin{matrix}(d_{m})^{\frac{1}{2}} − f̃(x), & if x ∈ Ω_{m}, \\ 0, & if x ∈ Ω \ Ω_{m}. \end{matrix}

Then ∥h̃_{m}(x)∥_{(M)} → 0, but |G(f̃(x) + h̃_{m}(x)) − G(f̃(x))| ≥ [ch|d_{m}| − |d_{m}| − 1]μ(Ω_{m})−Dμ(Ω_{m}) → \frac{1}{2}. Thus G is not Frechet-differentiable in (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{M}).

REFERENCES

[1] M. A. Krasnoselskii, Y. B. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd, Groningen 1961. (Translated from the first Russian edition by L. F. Boron).

[2] M. F. Sukhinin, Lower semi-Taylor mapping and sufficient conditions for an extremum, Math. USSR Sbornik, 73, 1 (1992), 257–271.

(received 20.01.2001)

Université Mohamed I, Faculté des Sciences, Département de Mathématiques et Informatique, Oujda, Maroc E-mail: lahrech@sciences.univ-oujda.ac.ma, addou@sciences.univ-oujda.ac.ma