The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem inf ess sup(Ω) , when the supremand f is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand f are also investigated.
RésuméDans cet article onétudie des conditions nécessaires et suffisantes pour l'existence de solutions pour le problème de minimisation inf ess sup x∈Ω f (∇u(x)) : u ∈ u0 + W 1,∞ 0 (Ω) lorsque la fonction f n'est pas une fonction convexe par niveaux. La stratégie utilisée pour obtenir ces conditions est celle de comparer ce problème avec son problème relaxé. On obtient comme condition nécessaire et suffisante une inclusion différentielle sur la donnée au bord. Onétudie aussi plusieurs conditions de convexité.
It is studied the lower semicontinuity of functionals of the type Ω f (x, u, v, ∇u)dx with respect to the (W 1,1 × L p )-weak * topology. Moreover in absence of lower semicontinuity, it is also provided an integral representation in W 1,1 × L p for the lower semicontinuous envelope.We are interested in studying the lower semicontinuity and relaxation of (1) with respect to the L 1 -strong ×L p -weak convergence. Clearly, bounded sequences {u n } ⊂ W 1,1 (Ω; R n ) may converge in L 1 , up to a subsequence, to a BV function. In this paper we restrict our analysis to limits u which are in W 1,1 (Ω; R n ). Thus, our results can be considered as a step towards the study of relaxation in BV (Ω; R n ) × L p (Ω; R m ) of functionals (1).We will consider separately the cases 1 < p < ∞ and p = ∞. To this end we introduce for 1 < p < +∞ the functionalfor any pair (u, v) ∈ W 1,1 (Ω; R n ) × L p (Ω; R m ), and for p = ∞ the functional J ∞ (u, v) := inf lim inf J(u n , v n ) :
The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Since the difference quotient is based on shifting the function, it cannot be generalized to the variable exponent case. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the variable exponent Sobolev space.
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