Free boundaries subject to topological constraints

Figure 1.  Mathematica plot of the free boundary of the double hairpin solution $ H_a(z) $ for $ a = 1/4 $, $ a = 1 $ and $ a = 2 $. Note that $ z = x_1 + i x_2 $ and $ x_2 $ is the horizontal axis in the diagram

Figure 2.  An illustration of the three cases for the free boundary $ F(u)\cap B_r(z) $ of a solution $ u $ having simply connected positive phase $ \mathbb{D}^+(u) $

Figure 3.  Illustrating $ \Omega $ in Case 4 of the proof of Lemma 3.6, whose existence is ruled out

Figure 4.  The conformal diffeomorphism $ U_a = H_a + i \tilde{H}_a $ mapping the right half of the positive phase $ \mathcal{D}_a = \Omega_a \cap \{x_1>0\} $ onto the slit domain $ \mathcal{S}_a $

Figure 5.  Mathematica plot of the free boundary of the Scherk solution $ S_s(x_1, x_2) $ for asymptotic slopes $ s = 1/8 $, $ s = 1/2 $ and $ s = 7/8 $. Note that in the diagram $ x_2 $ is the horizontal axis

Figure 6.  Mapping the subdomain $ \mathcal{D}^{\text{BSS}}_s $ of the positive phase of the Scherk solution $ S_s $ conformally onto the strip $ \mathcal{S}_l $ under $ U_s^{\text{BSS}} = S_s + i \tilde{S}_s. $ Note that $ Q_{\pm} $ is a saddle point of $ S_s $ with $ Q_{\pm}A_{\pm} $ and $ Q_{\pm}E_{\pm} $ being a steepest descent and a steepest ascent path from $ Q_{\pm} $, respectively