A. Gachechiladze

Georgian Technical University Tbilisi, Georgia

A MAXIMUM PRINCIPLE IN THE VARIATIONAL INEQUALITIES

In the present paper a week maximum principle for a bilinear, elliptic, nonsymmetric and coercive form, with the L^{∞} coeficients is given. The result is applied to prove the maximum principle for the variational inequalities.

Let f ∈ L_{2}(Ω), h ∈ H^{1}(Ω), H^{1}(Ω) be the real Sobolev space, K_{1} = { v ∈ H^{1}(Ω) : v ≥ h}, and a(u, v) be the above-mentioned bilinear form. The problem is to find u ∈ K_{1}, such that

a(u, v − u) ≥ ∫_{Ω} f · (v − u)dx, ∀v ∈ K_{1}.  (1)

In some restrictions there is proved the equality es supu=es suph for this inequality. This fact give us an opportunity to locate the set of coincidence (u = h) for some concrete cases. Inequality (1) is also considered on the set K_{2} = {v ∈ H^{1}(Ω) : ϕ ≥ v ≥ ψ}, with ϕ, ψ ∈ H^{1}(Ω). Based on the above week maximum principle, for some cases there is proved the equalities es supu=es supψ and es infu=es infϕ.

With the help of the maximum principle, written for the variational inequality in the set K_{1}, we prove the following result

a(u − u_{λ}, Ψ) ≥ 0, ∀Ψ ∈ H^{1}(Ω), Ψ ≥ 0.  (2)

Here λ ∈ R^{+}, u and u_{λ} are the solutions of the inequality (1) in the sets K_{1} and K_{1}^{λ} = {v ∈ H^{1}(Ω) : v ≥ h − λ}.

Finally, inequality (1) is considered with the implicit obstacle, i.e., in the set K_{3} = {v ∈ H^{1}(Ω) : v ≥ h − a(u, Φ)}, with Φ ∈ H^{1}(Ω), Φ ≥ 0. With the help of (2), the unique solvability of this problem is proved.