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Free boundaries subject to topological constraints

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    * Corresponding author 

Supported in part by NSF Grant DMS 1500771, a Simons Fellowship, and Simons Foundation grant (601948, DJ). NK was partially supported by Proyecto FONDECYT Iniciación No. 11160981

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  • We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis Caffarelli on free boundaries and by striking results of T. Colding and W. Minicozzi concerning finitely connected, embedded, minimal surfaces. We review our earlier work on the simplest case, one-phase free boundaries in the plane in which the positive phase is simply connected. We also prove a new, purely topological, effective removable singularities theorem for free boundaries. At the same time, we formulate some open problems concerning the multiply connected case and make connections with the theory of minimal surfaces and semilinear variational problems.

    Mathematics Subject Classification: Primary: 35R35, 35B08, 35D40, 53A10; Secondary: 35J61.

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  • 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

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