We consider a functional I(u) = ∫Ωf(∇ u(x)) dx on u0 + W1,1(Ω). Under the assumption that f is just convex, we prove a new Comparison Principle, we improve and give a short proof of Cellina's Comparison result for a new class of minimizers. We then extend a local Lipschitz regularity result obtained recently by Clarke for a wider class of functions f and boundary data u0 satisfying a new one-sided Bounded Slope Condition. A relaxation result follows.

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 equivalen

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The variational approach to various problems of shape optimization arising in both solid mechanics and fluid dynamics leads to the problem of minimizing a non convex functional of the form (see [7] and [8])where Ω is an open and bounded subset of R 2 and h : [0, ∞) → R is the minimum between two convex parabolas having the same axis of symmetry. The lack of convexity of the function h prevents the application of the direct method of the Calculus of Variations to the functional J. Still, when Ω is an open ball in R N , such functional features a unique radially symmetric minimizer provided h is only assumed to be lower semicontinuous and superlinear (see [4]). For Ω a square in R 2 , the corresponding convexified minimum problem, i.e. the minimum problem for the functionalis studied in [7] and [8] both from an analytical and a numerical point of view.In particular, the uniqueness result of [8] and the numerical tests presented in both papers show that the minimum problem for the non convex functional J has no solutions. In nearly optimal configurations, homogenization occurs.Mathematics Subject Classification (1991): 49J45