Abstract
We study four problems, two in L 1 and two in L ∞ , whose analogues in L 2 are the familiar minimum principles which lead to the Laplace equation. One possibility is to be given the boundary value ϕ = g and to minimize |vψ| 1 or |v| ∞ ; the gradient at a point (x,y) in Ω is measured by |vΔ | 2 = ψ 2 x + ψ 2 y . In the other problems we are given a vector field v: Ω - R 2 , and minimize either |v w -v| 1 or |v w -v| ∞ In each case we use the duality theory of convex analysis to give equivalent statements of the problem, often with an interpretation in mechanics and often partly solved. Nevertheless some questions still remain open.
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