Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models (Q2790595)

The paper deals with the partial regularity of minimizers for singular functionals of the form: NEWLINE\[NEWLINEI(u)= \int_{U} F \left( u,Du\right)+f(u) \,dx, \quad u \in H^{1}(U;\mathbb R^k)NEWLINE\]NEWLINE with quasiconvex integrand \(F\) in the gradient variable and the function \(f\) blows up to infinity at the boundary of a bounded open set \(K \subset \mathbb R^k\).NEWLINENEWLINEMore precisely, the function \(f:\mathbb R^k\,\rightarrow [0,\infty]\) is non-negative, convex and smooth in \(K\) such that NEWLINE\[NEWLINEf(z)<\infty, \;z\in K \text{ and }f(z)=\infty, \;z\in \mathbb R^k-KNEWLINE\]NEWLINE and \(f(z)\rightarrow \infty\) as dist\((z, \partial K) \rightarrow 0\) for \(z \in K\). Moreover, \(F :(z,P) \in \mathbb R^k\times M^{k\times n}\rightarrow \mathbb R\) is a uniformly strictly quasi-convex function in \(P\) and satisfies the following growth and regularity assumptions: NEWLINE\[NEWLINE|D_{P}^{2} F| \leq C,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\gamma |P| ^2 \leq F(z,P)+C,\,\,\,NEWLINE\]NEWLINE NEWLINE\[NEWLINE|F(z,P)-F(\bar{z},P)| \leq C(1+|P| ^2)|z-\bar{z}|,NEWLINE\]NEWLINE with \(C,\gamma>0\).NEWLINENEWLINEThe authors prove that the minimizer of \(I\) in the class of admissible functions NEWLINE\[NEWLINEA= \left\{ u \in H^1(U; \mathbb R^k): \;u=g \text{ on } \partial U \text{ in the sense of the trace}\right\},NEWLINE\]NEWLINE which exists by standard arguments in the calculus of variations, is in \(C^{1,\alpha} (U-U_0)\), where \(U_0 \subset U\) is an open set such that \(|U-U_0|=0\).NEWLINENEWLINEThis kind of energy functionals arise in some models in nematic liquid crystal theory, recently proposed by \textit{J. M. Ball} and \textit{A. Majumdar} [``Nematic liquid crystals: from Maier-Saupe to a continuum theory'', Proceedings of the European Conference on Liquid Crystals, Colmar, France (2009)].