## Some remarks concerning quasiconvexity and strong convergence.(English)Zbl 0628.49011

Let I be an integral of the calculus of variations of the form $$I[v]=\int_{\Omega}F(Dv)dx$$, where $$\Omega$$ is an open, bounded and smooth set of $${\mathbb{R}}^ n$$, $$v\in W^{1,q}(\Omega;{\mathbb{R}}^ N)$$ $$(q>1)$$, and Dv is the $$n\times N$$ matrix of the gradient of v. The function $$F=F(P)$$ satisfies the growth condition $$0\leq F(P)\leq c(1+| P|^ q)$$ for some positive constant c and for all $$P\in {\mathbb{R}}^{nN}$$. Moreover, F is uniformly strictly quasiconvex, in the sense that $\int_{\Omega}(F(A)+\gamma | D\phi |^ q)dx\leq \int_{\Omega}F(A+D\phi)dx,$ for some positive constant $$\gamma$$ and for every $$\phi \in W_ 0^{1,q}(\Omega;{\mathbb{R}}^ N)$$, and $$A\in {\mathbb{R}}^{nN}.$$
The authors prove that, if $$u_ K$$ converges to u in the weak topology of $$W^{1,q}(\Omega;{\mathbb{R}}^ N)$$ and if $$I[u_ k]$$ converges to I[u], then $$u_ k$$ converges, as $$k\to +\infty$$, to u in the strong topology of $$W^{1,q}_{loc}(\Omega;{\mathbb{R}}^ N)$$.
Reviewer: P.Marcellini

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 26B25 Convexity of real functions of several variables, generalizations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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### References:

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