Line-energy Ginzburg-Landau models: zero-energy states. (English) Zbl 1072.35051

The authors study the limits \(m\) of sequences \(\{m_\epsilon\}_{\epsilon\downarrow 0}\) such that \(\lim_{\epsilon\downarrow 0} E_\epsilon(m_\epsilon)=0\) (\`\` zero energy states\'\') for the energy functional \[ E_\epsilon(m) = \epsilon\int_\Omega |\nabla m|^2 + {1\over\epsilon} \int_\Omega (1-|m|^2)^2 + {1\over\epsilon}\int_{\mathbb{R}^2}|\nabla^{-1}\text{div}\,m|^2 \] where \(\Omega\subset\mathbb{R}^2\) and \(m:\Omega\rightarrow\mathbb{R}^2\) is assumed to be extended to be zero outside \(\Omega\). Such problems arise in various physical situations, such as smectic liquid crystals, soft ferromagnetic films and in the gradient theory of phase transitions. They have been studied by W. Jin and R. V. Kohn [J. Nonlinear Sci. 10, No. 3, 355–390 (2000; Zbl 0973.49009)], L. Ambrosio, C. De Lellis, and C. Mantegazza [Calc. Var. Partial Differ. Equ. 9, No. 4, 327–355 (1999; Zbl 0960.49013)] and A. DeSimone, R. V. Kohn, S. Müller and F. Otto [Proc. R. Soc. Edinb., Sect. A, Math. 131, No. 4, 833–844 (2001; Zbl 0986.49009]. A characteristic property of these problems is that as \(\epsilon\rightarrow 0\), minimizers converge to vector fields with line singularities.
The authors show that if \(\Omega=\mathbb{R}^2\), then \(m\) is either constant or a vortex, that is, \(m(x)=\pm(x-{\mathcal O})^\perp/|x-{\mathcal O}|\) for some \({\mathcal O}\in\mathbb{R}^2\) where \(\perp\) denotes counterclockwise rotation by \(\pi/2\).
They also show that within a class of domains \(\Omega\neq\mathbb{R}^2\) that permit the existence of zero energy states, either \(\Omega\) is a disc and \(m\) is a vortex, or \(\Omega\) is a strip and \(m\) is constant.
The proof is based on specific entropies which lead to a kinetic formulation, and on a careful analysis of the corresponding weak solutions by the method of characteristics.


35B65 Smoothness and regularity of solutions to PDEs
35J60 Nonlinear elliptic equations
Full Text: EuDML


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