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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
##### Citations:
Zbl 0703.49001; Zbl 0609.49009
Full Text:
##### References:
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