Orienting Method for Obstacle Problems

This paper deals with obstacle problems of type minimize R Ω F (x, v,∇v) dx subject to v ∈ W (Ω), v ≥ r in Ω, v = g on ∂Ω where Ω ⊂ R is a bounded open set and r, g ∈ W (Ω) (1 ≤ p ≤ ∞). To state some sufficient criteria for determining parts of the coincidence set C(u) = {x ∈ Ω : u(x) = r(x)} and of the non-coincidence set N (u) = {x ∈ Ω : u(x) > r(x)} of the optimal solution u to this obstacle problem, optimal solutions to some particular auxiliary problems without obstacle minimize R e ΩF (x, v,∇v) dx subject to v ∈ Ke Ω,g̃ = {v ∈ W 1,p(e Ω) : v = g̃ on ∂e Ω} are used as orienting tool. For this purpose, we do not assume any coercive assumption, but only the uniqueness of the optimal solution to auxiliary problems, which is ensured if e.g. the performance index is strictly convex in Ke .