Weak lower semicontinuity of polyconvex integrals. (English) Zbl 0813.49017

Malý, Jan

Proc. R. Soc. Edinb., Sect. A 123, No. 4, 681-691 (1993).

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).

Reviewer: A.Hoffmann (Ilmenau)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
90C25	Convex programming
49J52	Nonsmooth analysis
74B20	Nonlinear elasticity

Keywords:

polyconvex integrals; integral functional; sequentially weak lower semicontinuity

Citations:

Zbl 0703.49001; Zbl 0609.49009

References:

[1]	DOI: 10.1016/0022-1236(84)90041-7 · Zbl 0549.46019 · doi:10.1016/0022-1236(84)90041-7
[2]	DOI: 10.1007/BF00279992 · Zbl 0368.73040 · doi:10.1007/BF00279992
[3]	Acerbi, J. Math. Pures Appl. 62 pp 371– (1983)
[4]	Dacorogna, Appl. Math. Sciences 78 (1989)
[5]	DOI: 10.1007/BF01040914 · Zbl 0176.03503 · doi:10.1007/BF01040914
[6]	Marcellini, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 pp 391– (1986) · Zbl 0609.49009 · doi:10.1016/S0294-1449(16)30379-1
[7]	Dacorogna, C.R. Acad. Sci. Paris Sér. I Math. 311 pp 393– (1990)
[8]	Zhang, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 pp 345– (1990) · Zbl 0717.49012 · doi:10.1016/S0294-1449(16)30296-7