Evans, L. C.; Gariepy, R. F. Some remarks concerning quasiconvexity and strong convergence. (English) Zbl 0628.49011 Proc. R. Soc. Edinb., Sect. A 106, 53-61 (1987). 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 Cited in 1 ReviewCited in 22 Documents 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 Keywords:strong convergence; minimizing sequences; integral of the calculus of variations; growth condition; uniformly strictly quasiconvex PDFBibTeX XMLCite \textit{L. C. Evans} and \textit{R. F. Gariepy}, Proc. R. Soc. Edinb., Sect. A, Math. 106, 53--61 (1987; Zbl 0628.49011) Full Text: DOI References: [1] DOI: 10.1080/03605308408820337 · Zbl 0545.49019 [2] Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803 [3] DOI: 10.1007/BF00251360 · Zbl 0627.49006 [4] DOI: 10.1007/BF00275731 · Zbl 0565.49010 [5] Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Non-linear Functionals (1982) · Zbl 0484.46041 [6] DOI: 10.1007/BF00279992 · Zbl 0368.73040 [7] Giorgi, Rend. Mat. 8 pp 277– (1975) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.