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Weak lower semicontinuity of polyconvex integrals. (English) Zbl 0813.49017
The functional \[ u(\cdot) \mapsto\int_ \Omega {\mathbf W}(\nabla u(x))dx,\quad \Omega\subset \mathbb{R}^ n,\tag{1} \] is looked for sequentially weak lower semicontinuity under the assumption that the integrand can be rewritten in the form \({\mathbf W}(\nabla u)= {\mathbf f}(\nabla u,\text{adj }\nabla u,\text{ det }\nabla u)\) where the function \((a,b,c)\mapsto f(a,b,c)\) [cf. B. Dacorogna: “Direct methods in the calculus of variation” (1989; Zbl 0703.49001)] is convex and non-negative. It was known so far [cf. P. Marcellini, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 341-409 (1986; Zbl 0609.49009)] that the mapping \(u(\cdot)\mapsto \int_ \Omega{\mathbf f}(\nabla u,\text{ adj }\nabla u,\text{ det }\nabla u)dx\) on \(W^{1,p}(\Omega, \mathbb{R}^ n)\) is sequentially weakly lower semicontinuous when \(p> n-1\) or \(p= n-1= 1\).
The author shows that for \(p= n - 1\) and the restriction to diffeomorphisms the statement remains valid (lim sup has to be replaced by lim inf). For \(p< n-1\) the statement is not valid also under the above restriction. He gives a counterexample for the special functional \(\int_ \Omega\text{det }\nabla u dx\). Conditions for the existence of minimizers are given for some lower semicontinuous hull of the above given functional (1).

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
90C25 Convex programming
49J52 Nonsmooth analysis
74B20 Nonlinear elasticity
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[1] DOI: 10.1016/0022-1236(84)90041-7 · Zbl 0549.46019
[2] DOI: 10.1007/BF00279992 · Zbl 0368.73040
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