Summary
We prove regularity of minimizers of the functional\(F(s,{\mathbf{ }}u) = \int\limits_\Omega {\left( {\kappa \left| {\nabla s} \right|^2 + s^2 \left| {\nabla u} \right|^2 + \Psi (s)} \right)} {\mathbf{ }}dx\) recently suggested by Ericksen [10] for the statics of nematic liquid crystals. We show that, given locally minimizing pairs (s, u),s has a continuous representative, ands, u are smooth outside the set {s=0}. The proof relies upon higher integrability estimates, monotonicity, and decay lemmas.
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Ambrosio, L. Regularity of solutions of a degenerate elliptic variational problem. Manuscripta Math 68, 309–326 (1990). https://doi.org/10.1007/BF02568766
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DOI: https://doi.org/10.1007/BF02568766