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Regularity for critical points of a non local energy. (English) Zbl 0889.58022

The author considers the regularity of critical points of an energy functional defined on \({\mathbf H}^1 (B^3,S^2)\) by \[ E(u)= {1\over 2} \int_{B^3} |\nabla u |^2 -{1\over 2} \int_{B^3} H\cdot u+ \int_{B^3} F(u) \] where \(H\in {\mathbf L}_2 (\mathbb{R}^3, \mathbb{R}^3)\) is a solution of a certain nonlocal problem and \(F\in {\mathcal C}^1 (\mathbb{R}^3, \mathbb{R})\) is a nonlinear term satisfying several growth restrictions. Problems of this sort arise in micromagnetism theory. The author proves that the critical points of \(E\) are smooth except in a subset of one-dimensional Hausdorff measure zero.
Reviewer: V.Moroz (Minsk)

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E20 Harmonic maps, etc.
35B65 Smoothness and regularity of solutions to PDEs
35J20 Variational methods for second-order elliptic equations
35Q60 PDEs in connection with optics and electromagnetic theory
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