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Quasiconvexity and partial regularity in the calculus of variations. (English) Zbl 0627.49006

The author proves the following regularity theorem: Let \(F: {\mathbb{R}}^{nN}\to {\mathbb{R}}\) be a \(C^ 2\)-function satisfying \(| D^ 2F| \leq\) constant. Let us assume also that F is uniformly strictly quasiconvex, in the sense that \[ \int_{B(x,r)}(F(\xi)+\gamma | D\phi (y)|^ 2)dy\leq \int_{B(x,r)}F(\xi +D\phi (y))dy, \] for some \(\gamma >0\) and for all \(\xi \in {\mathbb{R}}^{nN}\) and \(\phi \in C^ 1_ 0(B(x,r);{\mathbb{R}}^ N)\), for all x, r. Let \(u\in H^ 1(\Omega;{\mathbb{R}}^ N)\) be a minimize for the integral of F, i.e.: \[ \int_{\Omega}F(Du(x))dx\leq \int_{\Omega}F(Du(x)+D\phi (x))dx, \] for every \(\phi \in H^ 1_ 0(\Omega;{\mathbb{R}}^ N)\), where \(\Omega\) is an open bounded set of \({\mathbb{R}}^ n\). Then, there exists an open set \(\Omega_ 0\subset \Omega\) such that \(| \Omega \setminus \Omega_ 0| =0\) and \(Du\in C^{\alpha}(\Omega_ 0;{\mathbb{R}}^{nN})\) for each \(\alpha\in (0,1)\). Moreover, if \(F\in C^{\infty}({\mathbb{R}}^{nN})\) then \(u\in C^{\infty}(\Omega_ 0;{\mathbb{R}}^ N).\)
The author gives also an analogous theorem in the core of functions F with growth of order \(q\geq 2\). These interesting results extend to the quasiconvex core the partial regularity known in the convex case.
More recently, new extensions and proofs have been given by N. Fusco and J. Hutchinson [Manuscr. Math. 54, 121-143 (1986; Zbl 0587.49005)], M. Giaquinta and G. Modica [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 185-208 (1986; Zbl 0594.49004)] and the author and R. F. Gariepy [Indiana Univ. Math. J. 36, 361-371 (1987)].
Reviewer: P.Marcellini

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
26B25 Convexity of real functions of several variables, generalizations
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