Some properties of the solutions of the Ginzburg-Landau equation on an open set in R(2)

For a solution u of -Delta u = u(1-\u\(2)) on the whole plane, \u\ < 1 holds everywhere unless u = e(i alpha) for some alpha is an element of R; the derivatives of order k have moduli less than or equal to a constant M(k) depending only on k. For a solution u on an open set Omega not equal R(2), the moduli of u and its derivatives have upper bounds depending only on the distance to R(2)\Omega; therefore the set of solutions on a given Omega is compact in C(Omega) for the topology of uniform convergence on compact subsets of Omega. For a solution u such that \u\ < 1, 1 - \u\ satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions. Finally, if w is bounded and \u\ has a lim sup less than or equal to m at each boundary point, then \u\ less than or equal to m in w if m greater than or equal to 1, but if m < 1 then \u\ admits only a majorant S-m(w) with values in ]m, 1[ and sufficient conditions are given for lim S-m(w) = 0 or S-m(w) = O(m) as m --> 0.