Remarks on the Relaxation of Integrals of the Calculus of Variations

Abstract

If we want to minimize a functional of the type:

$${\rm{F}}\left( {{\rm{u}},\Omega } \right)\, = \,\int\limits_\Omega {{\rm{f}}\left( {{\rm{x,u}}\left( {\rm{x}} \right),{\rm{Du}}\left( {\rm{x}} \right)} \right){\rm{dx}}}$$

(1)

where to is an open set in ${\rm{F}}\left( {{\rm{u}},\Omega } \right)\, = \,\int\limits_\Omega {{\rm{f}}\left( {{\rm{x,u}}\left( {\rm{x}} \right),{\rm{Du}}\left( {\rm{x}} \right)} \right){\rm{dx}}}$ it is useful to know whether the integral F(u,Ω)is sequentially lower semicontinuous (slsc) with respect to a topology T on w^1,p such that there exists a T-compact minimizing sequence.