On some nonlocal variational problems

Chipot, M; Gangbo, W; Kawohl, B (2006). On some nonlocal variational problems. Analysis and Applications, 4(4):345-356.

Abstract

We study uniqueness and non uniqueness of minimizers of functionals involving nonlocal quantities. We give also conditions which lead to a lack of minimizers and we show how minimization on an infinite dimensional space reduces here to a minimization on ℝ. Among other things, we prove that uniqueness of minimizers of functionals of the form ∫Ω a(∫Ω gu dx)|∇u|2 dx - 2 ∫Ω fu dx is ensured if a > 0 and 1/a is strictly concave in the sense that (1/a)″ < 0 on (0, ∞).

Abstract

We study uniqueness and non uniqueness of minimizers of functionals involving nonlocal quantities. We give also conditions which lead to a lack of minimizers and we show how minimization on an infinite dimensional space reduces here to a minimization on ℝ. Among other things, we prove that uniqueness of minimizers of functionals of the form ∫Ω a(∫Ω gu dx)|∇u|2 dx - 2 ∫Ω fu dx is ensured if a > 0 and 1/a is strictly concave in the sense that (1/a)″ < 0 on (0, ∞).

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