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Regularity of quasiconvex envelopes. (English) Zbl 0972.49024

In this very interesting paper the following question is addressed: consider the variational integral (\(\Omega\) denoting a domain in \(\mathbb{R}^n\)) \[ I(u)= \int_\Omega F(x,u,\nabla u) dx\tag{2} \] such that for some \(p\in (1,\infty)\) the integrand \(F:\Omega\times \mathbb{R}^N\times \mathbb{R}^{nN}\to \mathbb{R}\) satisfies the growth condition \[ c_1|\xi|^p- c_2\leq F(x,v,\xi)\leq c_3(|\xi|^p+ 1).\tag{2} \] Then \(I(u)\) makes sense for functions \(u: \Omega\to\mathbb{R}^N\) of Sobolev class \(W^{1,p}\) but without further assumptions, \(I\) in general fails to be lower-semicontinuous w.r.t. the weak topology of \(W^{1,p}\). So one passes to the l.s.c. envelope \(\overline I\) of (1) which has the representation \[ \overline I(u)= \int_\Omega\overline F(x,u,\nabla u) dx,\tag{3} \] \(\overline F\) denoting the quasiconvex envelope of \(F\) w.r.t. the last variable. Now, in order to justify the Euler-Lagrange equation associated to the functional (3), it is necessary to know what differentiability properties can be expected to hold for \(\overline F\). Up to now this question has only been settled by deriving explicit formulas for \(\overline F\) in certain examples which is extremely difficult.
For integrands \(F\) which depend also on \(u\) and satisfy (2) smoothness of \(\overline F\) in general fails to hold even if \(F\) is of class \(C^\infty\), examples are presented in Section 5 of the paper. The main results of the paper concern the case “\(F(\nabla u)\)” (see Theorem A and B) for which it is shown that differentiability or Lipschitz continuity of \(F\) together with some weaker growth conditions than stated in (2) implies \(C^1\)-regularity of the quasiconvex envelope \(\overline F\). Also an important class of non-differentiable integrands \(F(\nabla u)\) is given (see Theorem C and 5.5) such that \(\overline F\) is of type \(C^{1,1}_{\text{loc}}\). In Section 4 of the paper the reader will find further comments on the behaviour of the polyconvex envelope of \(F\). It is one of the most interesting aspects of the paper that the authors clearify the connection between the smoothness of \(\overline F\) and the growth of \(F\).

MSC:

49N60 Regularity of solutions in optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
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