Estimation a priori des solutions faibles de certaines sistèmes non linéaires elliptiques

The authors present a new and simpler proof of the following theorem: If u is a weak solution of the system −Dβ {aαβ (x, u, ∇u)Dαu} = f (x, u, ∇u) on Ω, where u = (u1(x), · · · , uN (x)), x = (x1, · · · , xn) ∈ Ω, f = (f 1, · · · , f N ) is defined on Ω × RN × RnN , under the S-conditions, and if aM + a∗ < 2, then u is H¨olderian on Ω with an exponent σ ∈]0, 1[ that depends only on a, b, a∗, b∗, M, λ, μ, n and N . Moreover, there exists a number k depending on the same constants and on mes Ω, so that [u]σ,Ω′ ≤ kd−σ for every Ω′ ⊂⊂ Ω with d = dist(Ω′, ∂Ω); k is independent of mes Ω if b = b∗ = 0. u will be a weak solution on Ω of the considered system if u ∈ H1 2 ∩ L∞(Ω, RN ) and if it satisfies ∫ Ω aαβ (x, u, ∇u)Dαu · Dβ ϕ dx = ∫ Ω f (x, u, ∇u) · ϕ dx for every ϕ ∈ ◦ H1 2 ∩ L∞(Ω, RN ). The S-conditions are: Aαβ (x) = aαβ (x, u(x), ∇u(x)) and f i(x, u(x), ∇u(x)) are measurable functions so that |u|L∞(Ω) ≤ M , Aαβ (x)ξαξβ ≥ λ|ξ|2 for all x ∈ Ω and ξ ∈ Rn, |Aαβ (x)| ≤ μ on Ω, |f (x, u(x), p)| ≤ aQ(x, p) + b for all x ∈ Ω and p ∈ RnN with Q(x, p) = Aαβ (x)pi αpi β , u(x) · f (x, u(x), p) ≤ a∗Q(x, p) + b∗ for all (x, p) ∈ Ω × RnN , where λ, μ > 0, M, a, b, b∗ > 0 and a∗ are constants.