## Rigorous results for the Ginzburg-Landau equations associated to a superconducting film in the weak $$\kappa$$ limit.(English)Zbl 0864.35097

A one-dimensional functional $\varepsilon_d= \varepsilon_d(f,A;h)= \int^{d/2}_{-d/2} \bigl[ {\textstyle{1\over 2}}(f^2-1)^2- {\textstyle{1\over 2}}+\kappa^{-2} f'{}^2+ f^2A^2+ (A'-h)^2\bigr]dx \tag{1}$ is introduced to look for the non-normal state of a superconducting film. The functional $$\varepsilon_d$$ depends on the parameters $$d$$, $$h$$, $$\kappa$$ where $$d$$ is proportional to the width of the film, $$h$$ is proportional to the intensivity of the exterior magnetic field, $$\kappa$$ is the characteristic of the material. The pairs $$(f,A)$$ describing the different states of the superconducting film, are the local or global minimum of $$\varepsilon_d$$. The authors concentrate the study on the behaviour of the superheating field if $$\kappa$$ tends to zero.
The organization of the paper is as follows: Section 1 gives the definition of the superheating field. In Section 2 the basic properties of the Ginzburg-Landau equations are established. In Section 3 and Section 4 the problem of superheating field in a finite interval is analyzed and a priori estimates for the solutions of the Ginzburg-Landau equations are given. In Section 5 and Section 6 the existence of the superheating field is proved and some monotonicity assertions for a maximal solution in the case of the half space are derived. Section 7 is devoted to the proof of a priori estimates. In Section 8 an inequality of De Gennes is proved.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 82D55 Statistical mechanics of superconductors 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 35B40 Asymptotic behavior of solutions to PDEs 35Q60 PDEs in connection with optics and electromagnetic theory
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