We study the minimum problem for functionals of the form F(u)=∫If(x,u(x),u′(x))dx,where the integrandf:I×Rm×Rm→Ris not convex in the last variable. We providean existence result assuming that the lower convex envelopef=f(x,p,ξ)offsatisfiesa suitable affinity condition on the set on whichf>fand that the map pi→f(x,p,ξ)is monotone with respect to one single componentpiof the vectorp. We show that ourhypotheses are nearly optimal, providing in such a way an almost necessary and sufficientcondition for the existence of minimizers.
The minimum problem for one-dimensional non-semicontinuous functionals / Zagatti, Sandro. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 1432-0835. - 61:1(2022), pp. 1-19. [10.1007/s00526-021-02138-8]
The minimum problem for one-dimensional non-semicontinuous functionals
Sandro Zagatti
2022-01-01
Abstract
We study the minimum problem for functionals of the form F(u)=∫If(x,u(x),u′(x))dx,where the integrandf:I×Rm×Rm→Ris not convex in the last variable. We providean existence result assuming that the lower convex envelopef=f(x,p,ξ)offsatisfiesa suitable affinity condition on the set on whichf>fand that the map pi→f(x,p,ξ)is monotone with respect to one single componentpiof the vectorp. We show that ourhypotheses are nearly optimal, providing in such a way an almost necessary and sufficientcondition for the existence of minimizers.| File | Dimensione | Formato | |
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