Variational inequality of elliptic type and Lagrange multipliers in elastic-plastic torsion

Carmela Vitanza

Dipartimento di Matematica - Università di Messina

Let Ω ⊂ lR^{n} be a bounded open set of C^{1} class and let K and K̃ be closed convex non empty subsets of H_{0}^{1}(Ω):

K = \{v ∈ H_{0}^{1}(Ω) : v ≥ 0 , ∑_{i=1}^{n} (\frac{∂v}{∂x_{i}} )^{2} ≤ 1 a. e. in Ω\}  (1)

K̃ = {v ∈ H_{0}^{1}(Ω) : v ≥ 0 , v(x) ≤ δ(x)}  (2)

Let a(u, v) the linear form

a(u, v) = ∫_{Ω} \{ ∑_{i,j=1}^{n} a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} + ∑_{i=1}^{n} d_{i} \frac{∂u}{∂x_{i}} v + ∑_{i=1}^{n} b_{i}u\frac{∂v}{∂x_{i}} + cuv\} dx (3)

which is assumed to be coercive and with a_{ij} ∈ C^{1}.

We show that if p > n and ∥f∥_{L^{p}} + ∥Lδ^{−}∥_{L^{p}} satisfies a smaller condition, the Variational Inequalities

Find u ∈ K : a(u, v − u) ≥ ∫_{Ω} f(v − u)dx ∀ v ∈ K  (4)

Find u ∈ K̃ : a(u, v − u) ≥ ∫_{Ω} f(v − u)dx ∀ v ∈ K̃  (5)

are equivalent. Such an equivalence allows us to prove the following theorems. Teorema 1 - Let us suppose that the assumptions (3) and (10) hold. Let u be the unique solutions to the problems (4) and (7). Then there exist two functions \overline{μ}, \overline{λ} ∈ L^{2}(Ω) such that

\{ \begin{matrix}\overline{μ}, \overline{λ} ≥ 0 \\ \overline{μ} u = 0 \\ \overline{λ}(δ(x) − u(x)) = 0 a. e. in Ω \\ Lu − f − \overline{μ} = −\overline{λ} \end{matrix}

Teorema 2 - Under the same assumptions of Theorem 1, the solution to Variational Inequality (4) verifies the conditions

\{ \begin{matrix}u(x) ≤ δ(x) \\ Lu − f − \overline{μ} =≤ 0 \\ (Lu − f − \overline{μ})(δ(x) − u(x)) = 0 \end{matrix}