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Lorentz space estimates and Jacobian convergence for the Ginzburg-Landau energy with applied magnetic field. (English) Zbl 1242.49030

In [S. Serfaty and I. Tice, J. Funct.Anal.254, No. 3, 773–825 (2008; Zbl 1159.26004)], the authors studied zeros of the complex-valued function \(u\) with possibly nonzero topological degrees (the so-called vortices), and found certain Marcinkiewicz-space (more precisely \(L^{2,\infty}\)) norm lower bounds for the difference between a certain total energy functional and the vortex-energy functional. The paper under review is a continuation of this earlier work. Here the full Ginzburg-Landau energy with an applied magnetic field \[ G_{\varepsilon}(u,A):= \frac12\int_{\Omega}|\nabla_Au|^2+|\mathrm{curl}\,A-h_{ex}|^2+\frac1{2\varepsilon^2}(1-|u|^2)^2 \] is studied, where \(h_{ex}\) is the intensity of the external magnetic field, \(\Omega\) is a bounded regular two-dimensional domain in the Euclidean space, \(u:\Omega\to\mathbb C\) is the order-parameter function, \(|u|^2\) represents the local density of superconducting electrons, \(A\) is the two-dimensional vector field which describes the vector-potential of the induced magnetic field, \(h=\text{curl} A\) and \(\nabla A\) is the covariant gradient of \(A\). The author focuses on certain extreme superconductors that correspond to states when \(\varepsilon\) is small. An interested reader could refer to [E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg-Landau model. Basel: Birkhäuser (2007; Zbl 1112.35002)] for more information about the topic.
The main results of the paper are contained in three principal theorems. The first of these is an improvement of Theorem 1.5 of [Sandier and Serfaty, loc. cit.] on \(\Gamma\)-convergence in the intermediate regime. Here, the author establishes sufficient conditions for a lower bound for the quantity \(G_{\varepsilon}(u_{\varepsilon},A_{\varepsilon})-f_{\varepsilon}(n)\) as long as \(n\) essentially lies within the interval \((1,h_{ex})\). The next main theorem is an application of the preceding one; it states a weak-* convergence of the vorticity measures (Jacobians) to either Dirac or probability measures, improving known compactness results. The final main result provides a two-sided estimate of the Marcinkiewicz norm of the gradient of a locally energy-minimizing solutions.
The proofs are deep and contain many results of independent separate interest. The techniques vary from algebraic tricks to rather deep knowledge from the theory of function spaces. For instance, in search of a space slightly larger than \(L^{2,\infty}\) but still smaller than \(W^{-1,p}\), the Lorentz-Zygmund spaces of the type \(L^{2,\infty;\gamma}\) with negative \(\gamma\) (studied for example in [C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces. Diss. Math. 175 (1980; Zbl 0456.46028)]), are involved.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
49S05 Variational principles of physics
35B25 Singular perturbations in context of PDEs
35J20 Variational methods for second-order elliptic equations
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References:

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