zbMATH — the first resource for mathematics

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:
  DOI: 10.1016/0022-1236(84)90041-7 · Zbl 0549.46019  DOI: 10.1007/BF00279992 · Zbl 0368.73040  Acerbi, J. Math. Pures Appl. 62 pp 371– (1983)  Dacorogna, Appl. Math. Sciences 78 (1989)  DOI: 10.1007/BF01040914 · Zbl 0176.03503  Marcellini, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 pp 391– (1986) · Zbl 0609.49009  Dacorogna, C.R. Acad. Sci. Paris Sér. I Math. 311 pp 393– (1990)  Zhang, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 pp 345– (1990) · Zbl 0717.49012
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.