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Lipschitz stability of strongly monotone variational inequalities

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a real Hilbert space (Hilbert space), let $K \subseteq H$ be nonempty, closed and convex, let $a$ be a bounded coercive bilinear form with constants $M, α > 0$ (Bounded, coercive and symmetric sesquilinear forms), and let $F_{1}, F_{2} \in H^{*}$ with the dual norm $∥ \cdot ∥_{*}$ (The dual space X^* of a normed space and its dual norm, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). Let $u_{i} \in K$ be the unique solution of the variational inequality with data $F_{i}$ , that is, $a (u_{i}, v - u_{i}) \geq F_{i} (v - u_{i})$ for every $v \in K$ (Stampacchia's variational inequality). Then $α ∥ u_{1} - u_{2} ∥ \leq ∥ F_{1} - F_{2} ∥_{*} .$

Facts & Assumptions

Given: A real Hilbert space $H$ , a nonempty closed convex $K \subseteq H$ , a bounded coercive bilinear form $a$ with constants $M, α > 0$ , bounded linear functionals $F_{1}, F_{2}$ on $H$ , and their unique variational solutions $u_{1}, u_{2} \in K$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice, consumed through the existence-and-uniqueness theorem for the variational inequality.

[F1]

Stampacchia's variational inequality: for each bounded linear functional $F$ on $H$ there is exactly one $u \in K$ with $a (u, v - u) \geq F (v - u)$ for every $v \in K$ ; in particular $u_{1}$ and $u_{2}$ are well defined and satisfy $a (u_{1}, v - u_{1}) \geq F_{1} (v - u_{1})$ and $a (u_{2}, v - u_{2}) \geq F_{2} (v - u_{2})$ for all $v \in K$ .

[F2]

Bounded, coercive and symmetric sesquilinear forms: $a$ is bilinear with $a (u, u) \geq α ∥ u ∥^{2}$ for every $u \in H$ .

[F3]

The dual space X^* of a normed space and its dual norm, The operator norm as the least bound and as the unit-sphere or unit-ball supremum: $H^{*}$ is the normed space of bounded linear functionals with dual norm $∥ G ∥_{*} = \sup {∣ G (w) ∣ : ∥ w ∥ \leq 1}$ , so $∣ G (w) ∣ \leq ∥ G ∥_{*} ∥ w ∥$ for every $w \in H$ ; in particular $F_{1} - F_{2} \in H^{*}$ .

Proof

technique · direct

Given: The setting above, with the unique solutions $u_{1}, u_{2} \in K$ of the two variational inequalities.

1.1givenA1F1

Testing the inequality for $u_{1}$ at the admissible point $v = u_{2}$ and the inequality for $u_{2}$ at $v = u_{1}$ [F1] gives $a (u_{1}, u_{2} - u_{1}) \geq F_{1} (u_{2} - u_{1})$ and $a (u_{2}, u_{1} - u_{2}) \geq F_{2} (u_{1} - u_{2})$ .

2.1step 1.1F2algebra

Adding the two inequalities of step 1.1 and using bilinearity [F2] gives $a (u_{1} - u_{2}, u_{2} - u_{1}) \geq (F_{1} - F_{2}) (u_{2} - u_{1})$ , that is, $- a (u_{1} - u_{2}, u_{1} - u_{2}) \geq (F_{1} - F_{2}) (u_{2} - u_{1})$ ; coercivity [F2] bounds the left-hand side by $- α ∥ u_{1} - u_{2} ∥^{2}$ , so $α ∥ u_{1} - u_{2} ∥^{2} \leq - (F_{1} - F_{2}) (u_{2} - u_{1}) = (F_{1} - F_{2}) (u_{1} - u_{2})$ .

3.1step 2.1A1F3algebra∎

If $u_{1} = u_{2}$ the asserted inequality is trivial. Otherwise the dual-norm estimate [F3] gives $(F_{1} - F_{2}) (u_{1} - u_{2}) \leq ∥ F_{1} - F_{2} ∥_{*} ∥ u_{1} - u_{2} ∥$ , so step 2.1 yields $α ∥ u_{1} - u_{2} ∥^{2} \leq ∥ F_{1} - F_{2} ∥_{*} ∥ u_{1} - u_{2} ∥$ ; dividing by the positive number $∥ u_{1} - u_{2} ∥$ gives $α ∥ u_{1} - u_{2} ∥ \leq ∥ F_{1} - F_{2} ∥_{*}$ , which is the assertion; Countable Choice was used only through the existence and uniqueness theorem [A1].

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Sources

Nguyen Dong Yen and Bui Trong Kim, Linear operators satisfying the assumptions of some generalized Lax-Milgram theorems, Acta Mathematica Vietnamica 26(3) (2001), 407-417 (standard reference, not scraped)
J. T. Oden and N. Kikuchi, Theory of variational inequalities with applications to problems of flow through porous media, International Journal of Engineering Science 18 (1980), 1173-1284 (standard reference, not scraped)