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Relaxation of multiple integral in subcritical Sobolev spaces. (English) Zbl 0915.49011
The paper deals with the relaxation of integral functionals $$F(u)=\int_\Omega W(\nabla u) dx$$ ($$\Omega\subset{\mathbb R}^N$$, $$u\in W^{1,N}(\Omega,{\mathbb R}^N)$$) in the case when there is a gap between the weak $$W^{1,p}$$ topology used for the relaxation (usually the choice of $$p$$ is due to a $$p$$-growth condition from below on $$W$$) and the growth from above, of order at most $$q$$. A typical example of this situation occurs when $$W(\xi)=f(\xi)+g(\text{det }\xi)$$ where $$f$$ has $$p$$-growth for some $$p\in (N-1,N)$$ and $$g$$ has linear growth (in this case $$q=N$$).
The main result of the paper, under technical assumptions which include the previous example, states that if $$p>q-1$$ the relaxed functional $${\mathcal F}_p(u)$$ is greater than $$\int_\Omega QW(\nabla u) dx$$ for any $$u\in W^{1,p}(\Omega,{\mathbb R}^N)$$, with equality if $$| \nabla u| \in L^N(\Omega)$$, where $$QW$$ is the quasi convex envelope of $$W$$. By a detailed analysis of an example it is shown that in general a strict inequality may hold if $$| \nabla u|$$ is not in $$L^N(\Omega)$$.
Reviewer: L.Ambrosio (Pisa)

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 74B20 Nonlinear elasticity
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##### References:
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