Degenerate elliptic systems and applications to Ginzburg-Landau type equations. I
The authors consider degenerate elliptic systems related with the Ginzburg-Landau type functional \[ E_\varepsilon(u, G)= {1\over n} \int_G \Biggl( |Du|^2+ {1\over 4\varepsilon^2} (1- |u|^2)^2\Biggr) dx,\tag{1} \] where \(G\) is a bounded smooth domain \(\mathbb{R}^n\). In the first part, the regularity and a priori estimates for solutions of \(p\)-harmonic systems \(\text{div}(a(x) |Du|^{p- 2} Du)= f(x)|Du|^p\) in \(B_2\), are proved with Hölder continuous coefficients \(a(x)\) and \(f(x)\). In the sequel, existence and stability inequalities for the functional (1) are proved, when instead of \(G\), the \((n- 1)\)-dimensional manifold \(S^{n- 1}\) or an \((n- 1)\)-dimensional compact Riemannian manifold without boundary are taken, as well as some applications in the study of the Ginzburg-Landau type functional. Finally in the third part, it is proved that from any sequence of minimizers of the higher-dimensional Ginzburg-Landau type functional (1) one can choose a subsequence strongly converging everywhere in \(G\) except a finite number of points, the limit of which is a weak solution to the problem \[ - \text{div}( |Du_\varepsilon |^{n- 2} Du_\varepsilon)= {1\over \varepsilon^n} (1- |u_\varepsilon |^2)^2 u_\varepsilon\quad \text{in}\quad G,\quad u_\varepsilon= g\quad\text{on} \quad \partial G. \]
higher-dimensional Ginzburg-Landau type functional
scientific article; zbMATH DE number 852859
scientific article