Variational inequalities for energy functionals with nonstandard growth conditions

Martin Fuchs and Li Gongbao

Abstract and Applied Analysis, 1998, vol. 3, 1-24

Abstract:

We consider the obstacle problem { minimize ? ? ? ? ? ? ? ? I ( u ) = ? O G ( ? u ) d x ? ? among functions ? ? u : O ? R such ? that ? ? ? ? ? ? ? u | ? O = 0 ? ? and ? ? u = F ? ? a .e . for a given function F ? C 2 ( O ¯ ) , F | ? O < 0 and a bounded Lipschitz domain O in R n . The growth properties of the convex integrand G are described in terms of a N -function A : [ 0 , 8 ) ? [ 0 , 8 ) with lim t ? 8 ¯ A ( t ) t - 2 < 8 . If n = 3 , we prove, under certain assumptions on G , C 1 , 8 -partial regularity for the solution to the above obstacle problem. For the special case where A ( t ) = t ln ( 1 + t ) we obtain C 1 , a -partial regularity when n = 4 . One of the main features of the paper is that we do not require any power growth of G .

Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:495907

DOI: 10.1155/S1085337598000438

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