Lobachevskii Journal of Mathematics Vol. 6, 2000, 19–32 \begin{matrix}c \\ ◯ \end{matrix}A.Addou, S.Lahrech

A.Addou, S.Lahrech

SUFFICIENT CONDITIONS FOR ELLIPTIC PROBLEM OF OPTIMAL CONTROL IN ℝ^{N} IN ORLICZ SOBOLEV SPACE

(submitted by A.V. Lapin)

Abstract. We consider here a problem for which we seek the local minimum in Orlicz Sobolev spaces (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{M}) for the Gâteaux functional J(f) ≡ ∫_{Ω} v(x, u, f)dx,where u is the solution of Dirichlet

problem with Laplacian operator associated to f and ∥.∥_{M} is the Orlicz norm. Note that, under the rapid growth conditions on v, the (G.f) J is not necesseraly Frechet differentiable in (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{M}). In this note, using a recent extension of Frechet Differentiability,(see [2]) ,we prove that, under the rapid growth conditons on v the (G.f) is differentiable for the new notion. Thus we can give sufficient conditions for local minimum.

1. Preliminaries

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

Let Ω be an open subset of ℝ^{n} ,with Lebesgue measure, and let M be a N function(i.e a real value 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) realvalue measurable functions u on Ω such that ∫_{Ω} M(u(x))dx < +∞, and the

Orlicz space L_{M}^{∗}(Ω) as the lineaire hull of L_{M}(Ω). We say that a sequence of functions u_{n}(x) ∈ L_{M}^{∗}(Ω), (n = 1, 2.....) is mean convergent to the function u_{0}(x) ∈ L_{M}^{∗}(Ω) if lim_{n→+∞} ∫_{Ω} M(u_{n}(x) − u_{0}(x))dx = 0. Every sequence u_{n}(x)

which converges in the norm of L_{M}^{∗}(Ω) to some function u_{0}(x) is also mean convergent to u_{0}(x). We shall say that the N-function M_{1}(u) increases essentialy

more rapidly than the N-function M(u) if,for arbitrary λ > 0, \frac{M(λu)}{M_{1}(u)} → 0 as u → +∞. 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 re-

spect to the Orlicz norm: ∥u∥_{M} = sup\{|∫_{Ω} u(x)v(x)dx| : ∫_{Ω} \overline{M}(v(x))dx ≤ 1\}.

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

D-finition 1.1. 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{Q}).

Suppose that Q is sufficiently smooth and bounded domain in ℝ^{n}. Let us consider the problem :

\{ \begin{matrix}Au = f, & (1.1) \\ u/_{∂Q} = 0 & (1.2) \end{matrix}

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

Theorem 1.1. [Agmon-Douglis-Niremberg] Let 1 < q < ∞, then we have ∀f ∈ L^{q}(Q), ∃ u ∈ W^{2,q}(Q) ⋂ W_{0}^{1,q}(Q) unique solution of problem (1.1),(1.2). Moreover ∀m ≥ 0 if f ∈ W^{m,q}(Q),then u ∈ W^{m+2,q}(Q) and ∥u∥_{W^{m+2,q}(Q)} ≤ c∥f∥_{W^{m,q}(Q)}.

Let f ∈ F ⊂ W_{0}^{1,2}(Q) a control. Let us consider J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx +

c_{k}∥f∥_{W^{1,2}(Q)}^{2}, (k = 0, 1, 2, ..., s_{1}) and J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx , (k = s_{1} + 1, s_{1} +

2, ..., s_{1}+s_{2}), where u is a solution of problem (1.1),(1.2) in W_{0}^{1,2}(Q) ⋂ W^{2,2}(Q). We consider tree 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)

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

1.4. Taylor mapping and lower semi-Taylor mapping. Let ∥.∥_{W^{1}L_{M}^{∗} (Ω)} the usual norm in W_{0}^{1}L_{M}^{∗}(Ω), F subset of W_{0}^{1}L_{M}^{∗}(Ω), τ a topology in F, Y a norm space with a norm ∥.∥_{Y}. According to [2] a map r : F ⟶ Y, (respectively r : F ⟶ ℝ), is called (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) infinitely small, (respectively (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) lower infinitely semi small), 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}}),

where O_{f} is the neighbourhood of f in (F, τ). A map J : F → Y,(respectively J : F → ℝ), is called (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) Taylor mapping,(respectively (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) lower semi-Taylor mapping),of order p_{1} at f ∈ F if they exists a such symmetrical k linear maps J^{(k)}(f) : W_{0}^{1}L_{M}^{∗}(Ω) → Y (not necesseraly continuous), (respectively J^{(k)}(f) : W_{0}^{1}L_{M}^{∗}(Ω) → ℝ (k = 1, .., p_{1})) and a neighbourhood O_{f} ∈ τ: ∀ h ∈ W_{0}^{1}L_{M}^{∗}(Ω) we have :

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 is (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) infinitely small (respectively (τ, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) lower infinitely semi small), of order p_{1} at f ∈ F. Let us remark that (J^{(1)}(f), .....J^{(p_{1})}(f)) are not unique. Let us solve the problems (1.3),(1.4)and (1.5). For the problem (1.5) let us introduce the lagrangian functions: L(f, y^{∗}, λ, λ_{0}) =

∑_{k=0}^{s_{1}} λ_{k}J_{k}(f)+ < y^{∗}, J(f) >, (1.5)

L_{f}(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}^{(1)}(f)+ < y^{∗}, J^{(1)}(f) >, (1.6)

L_{ff}(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}^{(2)}(f)+ < y^{∗}, J^{(2)}(f) >, (1.7)

where λ_{0} ∈ ℝ, y^{∗} ∈ (ℝ^{s_{2}})^{∗},λ ∈ (ℝ^{s_{1}})^{∗}. Also for the problem (1.4), let us introduce the lagrangian functions : L(f, y^{∗}, λ_{0}) = λ_{0}J_{0}(f)+ < y^{∗}, J(f) >, (1.8) L_{f}(f, y^{∗}, λ_{0}) = λ_{0}J_{0}^{(1)}(f)+ < y^{∗}, J^{(1)}(f) >, (1.9) L_{ff}(f, y^{∗}, λ_{0}) = λ_{0}J_{0}^{(2)}(f)+ < y^{∗}, J^{(2)}(f) >, (1.10) where λ_{0} ∈ ℝ, y^{∗} ∈ (ℝ^{s_{2}})^{∗}. Let us give the following two theorems 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 < ∞.

Theorem 1.3. Suppose the N-function M_{1}(u) increases essentialy more rapidly than the N-function M(u).Suppose also that the sequence u_{n}(x) ∈ L_{M_{1}}(Ω) is mean convergent to 0.Then u_{n}(x) converges to 0 with respect to the norm of the space L_{M}^{∗}(Ω): lim_{n→+∞} ∥u_{n}(x)∥_{M} = 0.

Let us give also the following lemma where the proof can be traced back to Lemma 1.1. Let (Ω, ∑, μ) a measurable space with a finite measure σ and X a complete metric linear space in M(Ω). By M(Ω) we denote the set of a measurable functions and almost every where finite x : Ω ⟶ ℝ with a metrizable topology τ(mes). Suppose that ∀ x ∈ X | x |∈ X and ρ(x, 0) = ρ(| x | , 0) ∀ x ∈ X. Then, ∀ x_{n} → 0 in X , ∃ (x_{n_{k}}) − subsequence of (x_{n}) and z ∈ X :| x_{n_{k}}(s) |≤ z(s) almost everywhere in Ω.

Using the same argument of 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 ∀u_{n} → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}), ∃ (u_{n_{k}}) − subsequence of (u_{n}) and z ∈ L_{M}^{∗}(Ω): | u_{n_{k}}(s) |≤ z(s) almost everywhere in Ω. Moreover 2z ∈ L_{M}(Ω) i.e ∫_{Ω} M(2z(x))dx < +∞.

Proof Let u_{n} → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}),then ∃(u_{n_{k}})−subsequence of (u_{n}) such that ∥u_{n_{k}}∥_{M} ≤ \frac{1}{2^{k}}. Let us put S_{n}(x) = ∑_{k=0}^{n} ∥u_{n_{k}}∥_{M}. So S_{n}(x) is convergent in

itself, i.e lim_{n,m→+∞} ∥S_{n}(x) − S_{m}(x)∥_{M} = 0. In fact we can suppose m > n,Thus

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

≤ ∑_{k=n+1}^{m} ∥u_{n_{k}}(x)∥_{M}

≤ ∑_{k=n+1}^{m} \frac{1}{2^{k}} → 0, as n, m → +∞.

This means that S_{n}(x) is convergent in itself, and since (L_{M}^{∗}(Ω), ∥.∥_{M}) is a complete space then S_{n}(x) converge in norm to S_{0}(x) ∈ L_{M}^{∗}(Ω) consequently, ∃(S_{p_{l}}) − subsequence of (S_{n}) such that S_{p_{l}} → S_{0}(x) almost everywhere in Ω. In fact, since S_{n}(x) converge in norm to S_{0}(x) ∈ L_{M}^{∗}(Ω) then ∫_{Ω} M(S_{n}(x) − S_{0}(x))dx → 0, and since M is continous in ℝ^{+} then ∃(S_{p_{l}}) −

subsequence of (S_{n}) such that S_{p_{l}} → S_{0}(x) almost everywhere in Ω. There-

fore ∃l_{0} ∈ ℕ, ∥ ∑_{k=p_{l_{0}}}^{+∞} |u_{n_{k}}(x)|∥_{M} < \frac{1}{2}. Let us put Z(x) = ∑_{k≥p_{l_{0}}}^{∗} |u_{n_{k}}(x)|, where

∑_{k≥p_{l_{0}}}^{∗} |u_{n_{k}}(x)| = lim_{l→+∞} ∑_{k=p_{l_{0}}}^{p_{l}} |u_{n_{k}}(x)|. So Z(x) = S_{0}(x)− ∑_{k=0}^{p_{l_{0}}−1} |u_{n_{k}}| ∈ L_{M}^{∗}(Ω),where

∥Z(x)∥_{M} < \frac{1}{2}. Let us put χ_{l}(x) = u_{n_{p_{l}}}(x) for l ≥ l_{0},so we obtain: forall l ≥ l_{0} |χ_{l}(x)| ≤ |Z(x)| almost everywhere in Ω, where (χ_{l}(x))_{l≥l_{0}} is subsequence of u_{n}. On the other hand , using the theorem(1.2), we obtain ∫_{Ω} M(2z(x))dx < +∞.

Thus we acheve the proof.

Lemma 1.3. Let Ω a bounded domain in ℝ^{n},and let M 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 solution of problem (1.1),(1.2) in C^{2}(Ω̄).

Proof Let f ∈ W_{0}^{1}L_{M}^{∗}(Ω).Since |t|^{p} ≤ M(t) then f ∈ L^{p}(Ω). Thus ∃ u ∈ W_{0}^{1,p}(Ω) unique solution 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^{2}(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)}.

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

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

Let Ω a bounded domain in ℝ^{n} and F subspace of W_{0}^{1}L_{M}^{∗}(Ω). Let G a functional defined by: G(f) = ∫_{Ω} v(x, u(x), f(x))dx , where u(x) is a solution

of problem (1.1),(1.2) in W_{0}^{1}L_{M}^{∗}(Ω) and v : Ω × ℝ × ℝ → ℝ. By τ_{M} we denote the topology generated by the Orlicz norm ∥.∥_{M}.

Theorem 2.1. Suppose that with every requirements of paragraph (1) and (2) the following conditions are verified : v, v_{uf}^{(2)}, v_{fu}^{(2)} are continuous in Ω × ℝ × ℝ, n > 1. Let M an N-function such that |t|^{p} ≤ M(t),for t ≥ t_{0}. Where p > n and t_{0} > 0. Let us suppose also that

\begin{aligned}| 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), \end{aligned}

where b_{5} ∈ L^{1}(Ω), b_{6} ∈ L^{1}(Ω). Then G is (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) Taylor mapping of first and second order at every 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,so since W_{0}^{1}L_{M}^{∗}(Ω) ↪ C(Ω̄) then f ∈ C(Ω̄), thus u ∈ C^{2}(Ω̄). Therefore ∫_{Ω} M(u(x))dx < +∞ and ∫_{Ω} M(f(x))dx < +∞.

24 A.ADDOU, S.LAHRECH Thus the functional G is finite.

\begin{aligned}Let R : & W_{0}^{1}L_{M}^{∗}(Ω) & ⟶ & W_{0}^{1}L_{M}^{∗}(Ω) \\ & h & ⟼ & (R(h))(x), \end{aligned}

where (R(h))(x) is a solution of problem

\{ \begin{matrix}Au = h, & (2.1) \\ u/_{∂Ω} = 0. & (2.2) \end{matrix}

A such solution exists ∀h ∈ W_{0}^{1}L_{M}^{∗}(Ω). Let G^{(1)}(f) 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)hdρ]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)]hdρ

+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.

Therefore: G^{(1)}(f) = ∫_{Ω} v_{u}^{(1)}(x, u, f)R(h)dx + ∫_{Ω} v_{f}^{(1)}(x, u, f)hdx. (2.3)

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. (2.4)

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

|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 (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) Taylor mapping, where τ_{M} is atopology generated by the Orlicz norm ∥.∥_{M}. Let f ∈ F, and let us prove that :

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)}) infinitely small of second order at zero. Assume the contrary, then ∃(h̃_{m})_{m∈ℕ} ∈ F and ε > 0 such that h̃_{m} → 0 in L_{M}^{∗}(Ω) and |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),the regularity of solution of the problem (1.1),(1.2), and the theorem (1.3),we obtain R(h̃_{m}) → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}), and using the lemma (1.2), we deduce that ∃z̃(x) ∈ L_{M}(Ω) : |(R(h̃_{m}))(x)| ≤ z̃(x), where 2z̃(x) ∈ L_{M}(Ω). Analogously, for h̃_{m} ∈ W_{0}^{1}L_{M}^{∗}(Ω), we obtain |h̃_{m}(x)| ≤ z̃_{1}(x), where 2z̃_{1}(x) ∈ 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)]dθdx

+ ∫_{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}]dλdx

− ∫_{Ω} ∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)hdλdx + ∫_{Ω} ∫_{0}^{1} ∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h)dλdθdx

= ∫_{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]hdλdx

− ∫_{Ω} ∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)hdλdx + ∫_{Ω} ∫_{0}^{1}∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h)dλdθdx.

Let A_{m}, B_{m} 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} 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})dθdx + ∫_{Ω} ∫_{0}^{1} B_{m}(x, λ)h̃_{m}^{2}dλdx

+ ∫_{Ω} ∫_{0}^{1} ∫_{0}^{1} F_{m}(x, θ, λ)R(h̃_{m})h̃_{m}dλdθdx|.

28 A.ADDOU, S.LAHRECH So,we obtain

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

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

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

≤ c_{5}[∫_{0}^{1} ∫_{Ω} |A_{m}(x, θ)|dxdθ + ∫_{0}^{1} ∫_{Ω} |B_{m}(x, λ)|dxdλ

+ ∫_{0}^{1} ∫_{0}^{1} ∫_{Ω} |F_{m}(x, θ, λ)|dxdλdθ]∥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)|dxdθ → 0 m → +∞,

∫_{0}^{1} ∫_{Ω} |B_{m}(x, λ)|dxdλ → 0 , m → +∞,

∫_{0}^{1} ∫_{0}^{1} ∫_{Ω} |F_{m}(x, θ, λ)|dxdλdθ → 0 , m → +∞;

which contradicts the assumption (2.3). Now let us prove that the functional G is (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) Taylor mapping of first order at every f ∈ F. Let f ∈ F, and let us prove that the map r(h) ≡ G(f + h) − G(f) − G^{(1)}(f)h is (τ_{M}, ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) infinitely small of first order at zero. Assume the contrary, then ∃ h̃_{m} ∈ L_{M}^{∗}(Ω) such that h̃_{m} → 0 in L_{M}^{∗}(Ω) but |r(h̃_{m})| ≥ ε∥h̃_{m}∥_{W^{1}L_{M}^{∗} (Ω)}. Thus R(h̃_{m}) → 0 in (L_{M}^{∗}(Ω), ∥.∥_{M}) , as m → +∞. Using the lemma (1.2),we deduce that there exists z̃(x) in L_{M}^{∗}(Ω) and there exists z̃_{1}(x) in L_{M}^{∗}(Ω): ∀m ∈ ℕ |[R(h̃_{m})](x)| ≤ z̃(x) and |h̃_{m}(x)| ≤ z̃_{1}(x) almost everywhere in Ω.Moreover 2z̃(x) ∈ L_{M}(Ω) and 2z̃_{1}(x) ∈ L_{M}(Ω). On the other hand

|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)

and

|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).

Using the dominated convergence theorem , we conclude that

∫_{Ω} |A_{m}(x)|dx → 0,as m → +∞

and

∫_{Ω} |B_{m}(x)|dx → 0,as m → +∞,

which contradicts the assumption (2.4). Thus we acheve the proof.

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 the 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 mapping of first and second order at every 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 semi Taylor

mapping of first and second order at every 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 map from (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{W^{1}L_{M}^{∗} (Ω)}) ⟶ ℝ^{s_{2}}, ∃ ŷ^{∗} ∈ (ℝ^{s_{2}})^{∗}, ∃α > 0 L_{f}(f̂, ŷ^{∗}, 1) = 0 and ∀h ∈ kerJ^{(1)}(f̂) L_{ff}(f̂, ŷ^{∗}, 1)(h, h) ≥ 2α∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{2}, where L_{f}(f̂, ŷ^{∗}, 1) and L_{ff}(f̂, ŷ^{∗}, 1) are given by formula (1.9),(1.10). Then f̂ is τ_{M} strict local minimum.

Proof All conditions of theorem (1.5) in [2] are verified,so f̂ is τ_{M} strict local minimum.

Theorem 2.3. Suppose that in the 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 mapping of first and second order at every 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 semi Tay-

lor mapping of first and second order at every 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)}) ⟶ ℝ^{s_{2}}, ∃λ̂ ∈ (ℝ^{s_{1}})^{∗}, ∃ŷ^{∗} ∈ (ℝ^{s_{2}})^{∗}, ∃γ ≥ 0, ∃λ̂_{k} > 0, (k = 1, .., s_{1}) : L_{f}(f̂, ŷ^{∗}, λ̂, 1) = 0 and ∀h ∈ L L_{ff}(f̂, ŷ^{∗}, λ̂, 1)(h, h) ≥ 2γ∥h∥_{W^{1}L_{M}^{∗} (Ω)}^{2}, where L_{f}(f̂, ŷ^{∗}, λ̂, 1) and L_{ff}(f̂, ŷ^{∗}, λ̂, 1) are defined by formula (1.6) and (1.7). Then f̂ is τ_{M} strict local minimum.

Proof All conditions of theorem (1.6) in [2] are verified,so f̂ is τ_{M} strict local minimum.

Theorem 2.4. Suppose that in the 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 mapping of first and second order at every 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 semi Taylor

mapping of first and second order at every 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 τ_{M} strict local minimum.

Proof All conditions of theorem (1.4) in [2] are verified,so f̂ is τ_{M} strict local minimum.

Remark 2.1. Let us remark that in the theorem (2.1),the increasing conditions verified 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) = a[ch|f| − |f| − 1],

and put b_{0}(x) ≡ 0,where a ∈ ℝ, a > 0. Let us put M(t) = e^{|t|} − |t| − 1,and let d_{m} → +∞. Let f̃ ∈ W_{0}^{1}L_{M}^{∗}(Ω) and let h̃_{m}(x) 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} Let Ω_{m} ⊂ Ω such that μ(Ω_{m}) =

(e^{d_{m}} − d_{m} − 1)^{−1}.Then∥h̃_{m}(x)∥_{(M)} → 0 but |G(f̃(x) + h̃_{m}(x)) − G(f̃(x))| ≥

a[ch|d_{m}|−|d_{m}|−1] → \frac{1}{2}. Thus G is not Frechet differentiable in (W_{0}^{1}L_{M}^{∗}(Ω), ∥.∥_{M}).

References

[1] M.A. KRASNOSELSKII AND, Y.B.RUTICKII, convex functions and Orlicz space, 1961 P.NOORDHOFF LTD.-GRONINGEN-THE NETHERLANDS Translated from the first russian edition by LEO F.BORON.

[2] M.F. SUKHININ, Lower semi-Taylor mapping and sufficient conditions for an extremum. ,Math.USSR Sbornik.-1992-V.73, No.1.-P.257-271.

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

Received August, 18, 1999