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December  2019, 39(12): 7213-7248. doi: 10.3934/dcds.2019301

## Free boundaries subject to topological constraints

 1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA 2 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago 7820436, Chile

* Corresponding author

Received  January 2019 Revised  May 2019 Published  September 2019

Fund Project: 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.

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.

Citation: David Jerison, Nikola Kamburov. Free boundaries subject to topological constraints. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7213-7248. doi: 10.3934/dcds.2019301
##### References:
 [1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar [2] H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461.  doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar [3] G. Baker, P. Saffman and J. Sheffield, Structure of a linear array of hollow vortices of finite cross-section, Journal of Fluid Mechanics, 74 (1976), 469-476.  doi: 10.1017/S0022112076001894.  Google Scholar [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure and Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar [5] L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, vol. 68 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068.  Google Scholar [6] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅰ. Lipschitz free boundaries are $C^{1, \alpha}$, Rev. Mat. Iberoamericana, 3 (1987), 139-162.  doi: 10.4171/RMI/47.  Google Scholar [7] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅲ. Existence theory, compactness, and dependence on $X$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1988), 583–602 (1989).  Google Scholar [8] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅱ. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math., 42 (1989), 55-78.  doi: 10.1002/cpa.3160420105.  Google Scholar [9] O. Chodosh and D. Maximo, On the topology and index of minimal surfaces, Journal of Differential Geometry, 104 (2016), 399-418.  doi: 10.4310/jdg/1478138547.  Google Scholar [10] H. I. Choi and R. Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Inventiones Mathematicae, 81 (1985), 387-394.  doi: 10.1007/BF01388577.  Google Scholar [11] T. H. Colding and W. P. Minicozzi Ⅱ, The Calabi-Yau conjectures for embedded surfaces, Annals of Mathematics, 167 (2008), 211-243.  doi: 10.4007/annals.2008.167.211.  Google Scholar [12] T. H. Colding and W. P. Minicozzi Ⅱ, On the structure of embedded minimal annuli, International Mathematics Research Notices, 2002 (2002), 1539-1552.  doi: 10.1155/S1073792802112128.  Google Scholar [13] T. H. Colding and W. P. Minicozzi Ⅱ, The space of embedded minimal surfaces of fixed genus in a 3-manifold. Ⅲ. Planar domains, Ann. of Math. (2), 160 (2004), 523–572. doi: 10.4007/annals.2004.160.523.  Google Scholar [14] T. H. Colding and W. P. Minicozzi Ⅱ, The space of embedded minimal surfaces of fixed genus in a 3-manifold. Ⅳ. Locally simply connected, Ann. of Math. (2), 160 (2004), 573–615. doi: 10.4007/annals.2004.160.573.  Google Scholar [15] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Archive for Rational Mechanics and Analysis, 195 (2010), 1025-1058.  doi: 10.1007/s00205-009-0227-8.  Google Scholar [16] L. Hauswirth, F. Hélein and F. Pacard, On an overdetermined elliptic problem, Pacific J. Math., 250 (2011), 319-334.  doi: 10.2140/pjm.2011.250.319.  Google Scholar [17] D. Jerison and N. Kamburov, Structure of one-phase free boundaries in the plane, International Mathematics Research Notices, 2016 (2016), 5922-5987.  doi: 10.1093/imrn/rnv339.  Google Scholar [18] D. Jerison and K. Perera, Higher critical points in an elliptic free boundary problem, J. Geom Anal., 28 (2018), 1258-1294.  doi: 10.1007/s12220-017-9862-8.  Google Scholar [19] D. Khavinson, E. Lundberg and R. Teodorescu, An overdetermined problem in potential theory, Pacific J. Math., 265 (2013), 85-111.  doi: 10.2140/pjm.2013.265.85.  Google Scholar [20] Y. Liu, K. Wang and J. Wei, On one phase free boundary problem in $\mathbb{R}^n$, preprint, arXiv: 1705.07345. Google Scholar [21] Y. Liu, K. Wang and J. Wei, Half space theorem for the Allen-Cahn equation, preprint, arXiv: 1901.07671. Google Scholar [22] W. H. Meeks Ⅲ and H. Rosenberg, The uniqueness of the helicoid, Annals of Mathematics, 161 (2005), 727-758.  doi: 10.4007/annals.2005.161.727.  Google Scholar [23] A. Ros, D. Ruiz and P. Sicbaldi, A rigidity result for overdetermined elliptic problems in the plane, Communications on Pure and Applied Mathematics, 70 (2017), 1223-1252.  doi: 10.1002/cpa.21696.  Google Scholar [24] A. Ros and P. Sicbaldi, Geometry and topology of some overdetermined elliptic problems, J. Differential Equations, 255 (2013), 951-977.  doi: 10.1016/j.jde.2013.04.027.  Google Scholar [25] R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, Journal of Differential Geometry, 18 (1983), 791-809.  doi: 10.4310/jdg/1214438183.  Google Scholar [26] M. Traizet, Classification of the solutions to an overdetermined elliptic problem in the plane, Geom. Funct. Anal., 24 (2014), 690-720.  doi: 10.1007/s00039-014-0268-5.  Google Scholar [27] K. Wang, The structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$, preprint, arXiv: 1506.00491. Google Scholar [28] K. Wang and J. Wei, On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg, Comm. Partial Differential Equations, 44 (2019), 837–858, arXiv: 1502.04680. doi: 10.1080/03605302.2019.1611846.  Google Scholar [29] K. Wang and J. Wei, Finite Morse index implies finite ends, Comm. Pure Appl. Math., 72 (2019), 1044–1119, arXiv: 1705.06831. doi: 10.1002/cpa.21812.  Google Scholar [30] G. S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem, Comm. Partial Differential Equations, 23 (1998), 439-455.  doi: 10.1080/03605309808821352.  Google Scholar [31] B. White, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals, Inventiones Mathematicae, 88 (1987), 243-256.  doi: 10.1007/BF01388908.  Google Scholar

show all references

##### References:
 [1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar [2] H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461.  doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar [3] G. Baker, P. Saffman and J. Sheffield, Structure of a linear array of hollow vortices of finite cross-section, Journal of Fluid Mechanics, 74 (1976), 469-476.  doi: 10.1017/S0022112076001894.  Google Scholar [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure and Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar [5] L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, vol. 68 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/gsm/068.  Google Scholar [6] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅰ. Lipschitz free boundaries are $C^{1, \alpha}$, Rev. Mat. Iberoamericana, 3 (1987), 139-162.  doi: 10.4171/RMI/47.  Google Scholar [7] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅲ. Existence theory, compactness, and dependence on $X$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 15 (1988), 583–602 (1989).  Google Scholar [8] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Ⅱ. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math., 42 (1989), 55-78.  doi: 10.1002/cpa.3160420105.  Google Scholar [9] O. Chodosh and D. Maximo, On the topology and index of minimal surfaces, Journal of Differential Geometry, 104 (2016), 399-418.  doi: 10.4310/jdg/1478138547.  Google Scholar [10] H. I. Choi and R. Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Inventiones Mathematicae, 81 (1985), 387-394.  doi: 10.1007/BF01388577.  Google Scholar [11] T. H. Colding and W. P. Minicozzi Ⅱ, The Calabi-Yau conjectures for embedded surfaces, Annals of Mathematics, 167 (2008), 211-243.  doi: 10.4007/annals.2008.167.211.  Google Scholar [12] T. H. Colding and W. P. Minicozzi Ⅱ, On the structure of embedded minimal annuli, International Mathematics Research Notices, 2002 (2002), 1539-1552.  doi: 10.1155/S1073792802112128.  Google Scholar [13] T. H. Colding and W. P. Minicozzi Ⅱ, The space of embedded minimal surfaces of fixed genus in a 3-manifold. Ⅲ. Planar domains, Ann. of Math. (2), 160 (2004), 523–572. doi: 10.4007/annals.2004.160.523.  Google Scholar [14] T. H. Colding and W. P. Minicozzi Ⅱ, The space of embedded minimal surfaces of fixed genus in a 3-manifold. Ⅳ. Locally simply connected, Ann. of Math. (2), 160 (2004), 573–615. doi: 10.4007/annals.2004.160.573.  Google Scholar [15] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Archive for Rational Mechanics and Analysis, 195 (2010), 1025-1058.  doi: 10.1007/s00205-009-0227-8.  Google Scholar [16] L. Hauswirth, F. Hélein and F. Pacard, On an overdetermined elliptic problem, Pacific J. Math., 250 (2011), 319-334.  doi: 10.2140/pjm.2011.250.319.  Google Scholar [17] D. Jerison and N. Kamburov, Structure of one-phase free boundaries in the plane, International Mathematics Research Notices, 2016 (2016), 5922-5987.  doi: 10.1093/imrn/rnv339.  Google Scholar [18] D. Jerison and K. Perera, Higher critical points in an elliptic free boundary problem, J. Geom Anal., 28 (2018), 1258-1294.  doi: 10.1007/s12220-017-9862-8.  Google Scholar [19] D. Khavinson, E. Lundberg and R. Teodorescu, An overdetermined problem in potential theory, Pacific J. Math., 265 (2013), 85-111.  doi: 10.2140/pjm.2013.265.85.  Google Scholar [20] Y. Liu, K. Wang and J. Wei, On one phase free boundary problem in $\mathbb{R}^n$, preprint, arXiv: 1705.07345. Google Scholar [21] Y. Liu, K. Wang and J. Wei, Half space theorem for the Allen-Cahn equation, preprint, arXiv: 1901.07671. Google Scholar [22] W. H. Meeks Ⅲ and H. Rosenberg, The uniqueness of the helicoid, Annals of Mathematics, 161 (2005), 727-758.  doi: 10.4007/annals.2005.161.727.  Google Scholar [23] A. Ros, D. Ruiz and P. Sicbaldi, A rigidity result for overdetermined elliptic problems in the plane, Communications on Pure and Applied Mathematics, 70 (2017), 1223-1252.  doi: 10.1002/cpa.21696.  Google Scholar [24] A. Ros and P. Sicbaldi, Geometry and topology of some overdetermined elliptic problems, J. Differential Equations, 255 (2013), 951-977.  doi: 10.1016/j.jde.2013.04.027.  Google Scholar [25] R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, Journal of Differential Geometry, 18 (1983), 791-809.  doi: 10.4310/jdg/1214438183.  Google Scholar [26] M. Traizet, Classification of the solutions to an overdetermined elliptic problem in the plane, Geom. Funct. Anal., 24 (2014), 690-720.  doi: 10.1007/s00039-014-0268-5.  Google Scholar [27] K. Wang, The structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$, preprint, arXiv: 1506.00491. Google Scholar [28] K. Wang and J. Wei, On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg, Comm. Partial Differential Equations, 44 (2019), 837–858, arXiv: 1502.04680. doi: 10.1080/03605302.2019.1611846.  Google Scholar [29] K. Wang and J. Wei, Finite Morse index implies finite ends, Comm. Pure Appl. Math., 72 (2019), 1044–1119, arXiv: 1705.06831. doi: 10.1002/cpa.21812.  Google Scholar [30] G. S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem, Comm. Partial Differential Equations, 23 (1998), 439-455.  doi: 10.1080/03605309808821352.  Google Scholar [31] B. White, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals, Inventiones Mathematicae, 88 (1987), 243-256.  doi: 10.1007/BF01388908.  Google Scholar
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
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 $\mathbb{D}^+(u)$
Illustrating $\Omega$ in Case 4 of the proof of Lemma 3.6, whose existence is ruled out
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 $\mathcal{S}_a$
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
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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