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NEW LOWER SEMICONTINUITY RESULTS FOR POLYCONVEX INTEGRALS
 

Summary: NEW LOWER SEMICONTINUITY RESULTS
FOR POLYCONVEX INTEGRALS
Emilio ACERBI
Gianni DAL MASO
Abstract
We study integral functionals of the form F (u, ) =

f(u) dx, defined for u
C1(; Rk), Rn . The function f is assumed to be polyconvex and to satisfy the
inequality f(A) c0|M(A)| for a suitable constant c0 > 0, where M(A) is the n-vector
whose components are the determinants of all minors of the k×n matrix A. We prove
that F is lower semicontinuous on C1(; Rk) with respect to the strong topology of
L1(; Rk). Then we consider the relaxed functional F , defined as the greatest lower
semicontinuous functional on L1(; Rk) which is less than or equal to F on C1(; Rk).
For every u BV (; Rk) we prove that F(u, )

f(u) dx + c0|Dsu|(), where
Du = u dx + Dsu is the Lebesgue decomposition of the Radon measure Du. More-
over, under suitable growth conditions on f , we show that F(u, ) =

  

Source: Acerbi, Emilio - Dipartimento di Matematica, Universitą degli Studi di Parma

 

Collections: Mathematics