An Integral Representation Result for the Γ-Limit of Functionals with Non-standard Growth Conditions in the Case of Elasticity

An integral representation result on regular functions is proved for the o -limit of a sequence of integral functionals defined in the vectorial case and modelled on elasticity theory functional Z z f (( x , e ( u )) dx where convex lagrangians satisfy a non-standard estimate $$ -c_{0} + c_{1} | \xi|^{\alpha }\leq f ( (x,\xi ) \leq c_{0} + c_{2} | \xi|^{\beta },\quad 1 \lt \alpha \leq \beta \lt \frac {n\alpha }{n-\alpha },\enskip c_{0}\geq 0,\enskip c_{1},c_{2} \gt 0. $$ When the limit integrand does not show Lavrent'ev phenomenon the representation result is also true on the whole space W 1, f ( z ; R n ).

Keywords:

• Γ-Convergence
• Integral Representation
• Lavrentiev Phenomenon