Volume 24, Number 4, October–December 2018
|Page(s)||1333 - 1343|
|Published online||20 November 2018|
Diagonal non-semicontinuous variational problems
265 – 34136
* Corresponding author: email@example.com
Accepted: 13 October 2017
defined on Sobolev spaces, where the integrand f:I×ℝm×ℝm→ℝ is assumed to be non convex in the last variable. Denoting by f̅ the lower convex envelope of f with respect to the last variable, we prove the existence of minimum points of F assuming that the application p↦f̅(⋅,p,⋅) is separately monotone with respect to each component pi of the vector p and that the Hessian matrix of the application ξ↦f̅(⋅,⋅,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.
Mathematics Subject Classification: 46B50 / 49J45
Key words: Non semicontinuous functional / minimum problem / Γ-convergence
© EDP Sciences, SMAI 2018
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