## 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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