We consider the following problems Minimize I(u) = Ω f(∇u(x)) + g(x, u(x))dz on u ∈ W 1,p 0 (Ω) : u− (x) ≤ u(x) ≤ u+ (x) Minimize I(u) = Ω f(∇u(x)) + g(x, u(x))dz on u ∈ W 1,∞ 0 (Ω) : ∇u(x) ∈ K where f : R n → R is a convex function, Ω is an open bounded subset of R, K is a closed convex subset of R n such that 0 ∈ int K and u− and u+ are suitable obstacles. We give conditions on the function g under which the two problems are equivalent
On the equivalence of two variational problems
TREU, GIULIA;
2000
Abstract
We consider the following problems Minimize I(u) = Ω f(∇u(x)) + g(x, u(x))dz on u ∈ W 1,p 0 (Ω) : u− (x) ≤ u(x) ≤ u+ (x) Minimize I(u) = Ω f(∇u(x)) + g(x, u(x))dz on u ∈ W 1,∞ 0 (Ω) : ∇u(x) ∈ K where f : R n → R is a convex function, Ω is an open bounded subset of R, K is a closed convex subset of R n such that 0 ∈ int K and u− and u+ are suitable obstacles. We give conditions on the function g under which the two problems are equivalent
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