Diagonal non-semicontinuous variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1333-1343.

We study the minimum problem for non sequentially weakly lower semicontinuos fucntionals of the form ( u ) = I f ( x , u ( x ) , u ' ( x ) ) d x , defined on Sobolev spaces, where the integrand f : I × m × m is assumed to be non convex in the last variable. Denoting by f ¯ the lower convex envelope of f with respect to the last variable, we prove the existence of minimum points of assuming that the application p f ¯ ( · , p , · ) is separately monotone with respect to each component p i of the vector p and that the Hessian matrix of the application ξ f ¯ ( · , · , ξ ) is diagonal. In the special case of functionals of sum type represented by integrands of the form f ( x , p , ξ ) = g ( x , ξ ) + h ( x , p ) , we assume that the separate monotonicity of the map p h ( , p ) , holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.

DOI : 10.1051/cocv/2017068
Classification : 46B50, 49J45
Mots clés : Non semicontinuous functional, minimum problem, Γ-convergence
Zagatti, Sandro 1

1
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Zagatti, Sandro. Diagonal non-semicontinuous variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1333-1343. doi : 10.1051/cocv/2017068. http://www.numdam.org/articles/10.1051/cocv/2017068/

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