A note on an overdetermined problem for minimal surface equation
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Abstract:
We show in this short note an overdetermined problem for minimal surface equation, which is defined over a domain with connected boundary, has a solution if and only if the domain is either the complement of a round disk or a half plane. Equivalently, we show that a minimal graph with connected zero boundary and makes a constant angle at all boundary points with a plane is a part of a catenoid or a half-plane.References
- H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math. 50 (1997), no. 11, 1089–1111. MR 1470317, DOI 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6
- Jaigyoung Choe, On the analytic reflection of a minimal surface, Pacific J. Math. 157 (1993), no. 1, 29–36. MR 1197043, DOI 10.2140/pjm.1993.157.29
- Tobias H. Colding and William P. Minicozzi II, Complete properly embedded minimal surfaces in $\mathbf R^3$, Duke Math. J. 107 (2001), no. 2, 421–426. MR 1823052, DOI 10.1215/S0012-7094-01-10726-6
- Tobias H. Colding and William P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211–243. MR 2373154, DOI 10.4007/annals.2008.167.211
- Manuel Del Pino, Frank Pacard, and Juncheng Wei, Serrin’s overdetermined problem and constant mean curvature surfaces, Duke Math. J. 164 (2015), no. 14, 2643–2722. MR 3417183, DOI 10.1215/00127094-3146710
- Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny, Minimal surfaces, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 339, Springer, Heidelberg, 2010. With assistance and contributions by A. Küster and R. Jakob. MR 2566897, DOI 10.1007/978-3-642-11698-8
- José M. Espinar and Jing Mao, Extremal domains on Hadamard manifolds, J. Differential Equations 265 (2018), no. 6, 2671–2707. MR 3804728, DOI 10.1016/j.jde.2018.04.044
- Mouhamed Moustapha Fall, Ignace Aristide Minlend, and Tobias Weth, Unbounded periodic solutions to Serrin’s overdetermined boundary value problem, Arch. Ration. Mech. Anal. 223 (2017), no. 2, 737–759. MR 3590664, DOI 10.1007/s00205-016-1044-5
- Laurent Hauswirth, Frédéric Hélein, and Frank Pacard, On an overdetermined elliptic problem, Pacific J. Math. 250 (2011), no. 2, 319–334. MR 2794602, DOI 10.2140/pjm.2011.250.319
- William H. Meeks III and Harold Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), no. 2, 727–758. MR 2153399, DOI 10.4007/annals.2005.161.727
- Johannes C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195–270. MR 173993, DOI 10.1090/S0002-9904-1965-11276-9
- Juncheol Pyo, Minimal annuli with constant contact angle along the planar boundaries, Geom. Dedicata 146 (2010), 159–164. MR 2644276, DOI 10.1007/s10711-009-9431-9
- Antonio Ros, David Ruiz, and Pieralberto Sicbaldi, A rigidity result for overdetermined elliptic problems in the plane, Comm. Pure Appl. Math. 70 (2017), no. 7, 1223–1252. MR 3666566, DOI 10.1002/cpa.21696
- Antonio Ros, David Ruiz, and Pieralberto Sicbaldi, Solutions to overdetermined elliptic problems in nontrivial exterior domains, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 1, 253–281. MR 4046014, DOI 10.4171/jems/921
- Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791–809 (1984). MR 730928
- James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
- Martin Traizet, Classification of the solutions to an overdetermined elliptic problem in the plane, Geom. Funct. Anal. 24 (2014), no. 2, 690–720. MR 3192039, DOI 10.1007/s00039-014-0268-5
- Martin Traizet, Hollow vortices and minimal surfaces, J. Math. Phys. 56 (2015), no. 8, 083101, 18. MR 3455335, DOI 10.1063/1.4927248
Additional Information
- Qing Cui
- Affiliation: School of Mathematics, Southwest Jiaotong University, 611756 Chengdu, Sichuan, People’s Republic of China
- Email: cuiqing@swjtu.edu.cn
- Received by editor(s): June 29, 2021
- Received by editor(s) in revised form: February 24, 2022
- Published electronically: June 30, 2022
- Additional Notes: The author was partially supported by National Natural Science Foundation of China (Grant No. 11601442) and Fundamental Research Funds for the Central Universities (Grant No. 2682020ZT103 and 2682021ZTPY043).
- Communicated by: Guofang Wei
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 5403-5409
- MSC (2020): Primary 35N25, 53A10, 53C24
- DOI: https://doi.org/10.1090/proc/16050