Quasiconvexity and partial regularity in the calculus of variations
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 \textit{N. Fusco} and \textit{J. Hutchinson} [Manuscr. Math. 54, 121-143 (1986; Zbl 0587.49005)], \textit{M. Giaquinta} and \textit{G. Modica} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 185-208 (1986; Zbl 0594.49004)] and the author and \textit{R. F. Gariepy} [Indiana Univ. Math. J. 36, 361-371 (1987)].
scientific article; zbMATH DE number 4019846
scientific article