Vortex-filaments pinning for inhomogeneous Ginzburg-Landau equations (Q2434834)

The paper studies the structure of vortices of the Ginzburg-Landau (GL) system with impurities in dimension \(N \geq 3\). In the framework of the GL theory, the energy of an inhomogeneous superconductor is modeled by the functional \(J(u)\) including the complex order parameter \(u\), which indicates the local state of the material and represents the density of superconducting electron pairs. The functional includes also a positive smooth function \(a(x)\) modeling the impurities in the sample whose values are temperature dependent (where \(x\) lies on the boundary \(\partial\Omega\) of the smooth domain). The minimizer \(u^\varepsilon\) is a weak solution of the corresponding GL equations. Earlier, it has been proved that in the case \(a(x)= 1\) the minimizers \(u^\varepsilon\) of the corresponding GL energy converge to a limit having a finite number of point singularities in the 2D case. The paper considers the case \(a(x)\neq 1\) at \(N\geq 3\). First, the monotonicity formula is derived. With this aim, the minimizers \(f^\varepsilon\) on \(\partial\Omega\) of the energy functional \(J(f^\varepsilon)\) are introduced. This minimizer \(f^\varepsilon\) is the unique positive solution of the corresponding equations. The real solution \(f^\varepsilon\) of these equations gives the density profile of vortex-free configurations. Then, the function \(v^\varepsilon\) as \(u^\varepsilon = f^\varepsilon v^\varepsilon\) is introduced, the corresponding equations are stated, and using maximum principles some bounds are defined. A computation directly gives the Pohozaev identity. For a minimizer \(u^\varepsilon\), the initial energy functional splits as \(J(u^\varepsilon) = J(f^\varepsilon) + E^\varepsilon(v)\) and for the new functional \(E^\varepsilon(v)\) an estimation is found. Then, it is shown that the \(\eta\)-compactness theorem bounds \(|v^\varepsilon|\) away from zero as soon as the local energy is bounded by \(\eta|\ln\varepsilon|\) with small \(\eta\) (where \(0 < \varepsilon < 1\) is a characteristic of the superconductor). Applying the Poincaré lemma, the Hodge-de Rham decomposition of \(v^\varepsilon\times dv^\varepsilon\) is deduced, and the \(L^2\) degree-norm of each of the three terms of this decomposition is estimated. Then, using the Radon-Nikodym theorem, the main theorem is formulated and proved for the minimizer \(u^\varepsilon\) and the function \(a(x)\) modeling the impurities causing the properties of pinning phenomena, limiting vortices and convergence. Namely, the authors state that the vortices are \((N-2)\)-rectifiable with zero mean curvature. Moreover, it is shown that vortices are forced to accumulate on the set of minima of \(a(x)\) defining the pinning phenomenon.