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The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal reference function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface. [ABSTRACT FROM AUTHOR]

In this paper we give a detailed asymptotic formula for the lowest eigenvalue of the magnetic Neumann Schrödinger operator in the ball in three dimensions with constant magnetic field, as the strength of the magnetic field tends to infinity. This asymptotic formula is used to prove that the eigenvalue is monotonically increasing for large values of the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper we study asymptotic behavior as ℇ→0 of Ginzburg–Landau functional Iℇ(v):=∫Ω(ℇ2v″2(s)+W(v′(s))+a(s)(v(s)+g(s))2) ds for v∈Hper2(Ω), where Ω⊆R is a bounded open interval, W is a non‐negative continuous function vanishing at ±1, a∈L1(Ω), and g is 1‐Lipschitz. Our consideration follows the approach introduced in the original paper by G. Alberti and S. Müller (Comm. Pure Appl. Math. 54 (2001), 761–825), where the case g=0 was studied. We show that their program can be modified in the case of functional Iℇ: we define suitable relaxation of Iℇ and prove a Γ‐convergence result in the topology of the so‐called Young measures on micropatterns. Moreover, we identify a unique minimizing measure for the functional in the limit, which is the unique translation‐invariant measure supported on the orbit of a particular periodic sawtooth function having minimal period and slope dependent on a derivative of g. [ABSTRACT FROM AUTHOR]

We consider a nonlinear homogenization problem for a Ginzburg–Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ℇ such that the average distances between the inclusions are of the order of ℇ, the inclusions sizes are of the order of ℇα, α>1, and the coefficient in front of the surface energy term, is of the order of ℇβ. We found that the parametric half‐plane {(α,β): α>1, -∞<β+∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential. We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems. [ABSTRACT FROM AUTHOR]

We study a nonlinear homogenization problem in perforated domains for the Ginzburg–Landau functional with a surface energy integral term which was introduced in the theory of liquid crystals. Under certain conditions between the inclusion size and the order of the surface energy integral we compute the nonlinear homogenized problem and prove the convergence of the solutions. An additional relation between the domain size and physical parameters provides the uniqueness of the homogenized limit. Our proof is based on a variational approach and does not require strict periodicity of the perforated domains. [ABSTRACT FROM AUTHOR]