A Hölder estimate for entire solutions to the two-valued minimal surface equation

Rosales, Leobardo

doi:10.1090/proc/12774

A Hölder estimate for entire solutions to the two-valued minimal surface equation

Author: Leobardo Rosales
Journal: Proc. Amer. Math. Soc. 144 (2016), 1209-1221
MSC (2010): Primary 49Q15
DOI: https://doi.org/10.1090/proc/12774
Published electronically: November 20, 2015
MathSciNet review: 3447673
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Abstract: We prove a Hölder estimate near infinity for solutions to the two-valued minimal surface equation over $\mathbb{R}^{2} \setminus \{0\},$ and give a Bernstein-type theorem in case the solution can be extended continuously across the origin. The main results follow by modifying methods used to study exterior solutions to equations of minimal surface type.

[1] William K. Allard, On the first variation of a varifold: boundary behavior, Ann. of Math. (2) 101 (1975), 418–446. MR 397520, https://doi.org/10.2307/1970934
[2] W. K. Allard and F. J. Almgren Jr., The structure of stationary one dimensional varifolds with positive density, Invent. Math. 34 (1976), no. 2, 83–97. MR 425741, https://doi.org/10.1007/BF01425476
[3] Robert Finn, New estimates for equations of minimal surface type, Arch. Rational Mech. Anal. 14 (1963), 337–375. MR 157096, https://doi.org/10.1007/BF00250712
[4] D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377. MR 1062966, https://doi.org/10.1007/BF01231506
[5] Leobardo Rosales, The geometric structure of solutions to the two-valued minimal surface equation, Calc. Var. Partial Differential Equations 39 (2010), no. 1-2, 59–84. MR 2659679, https://doi.org/10.1007/s00526-009-0301-y
[6] Leobardo Rosales, Discontinuous solutions to the two-valued minimal surface equation, Adv. Calc. Var. 4 (2011), no. 4, 363–395. MR 2844510, https://doi.org/10.1515/ACV.2011.005
[7] Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797. MR 634285, https://doi.org/10.1002/cpa.3160340603
[8] Leon Simon, Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J. 25 (1976), no. 9, 821–855. MR 412605, https://doi.org/10.1512/iumj.1976.25.25066
[9] Leon Simon, A Hölder estimate for quasiconformal maps between surfaces in Euclidean space, Acta Math. 139 (1977), no. 1-2, 19–51. MR 452746, https://doi.org/10.1007/BF02392233
[10] Leon Simon, Equations of mean curvature type in 2 independent variables, Pacific J. Math. 69 (1977), no. 1, 245–268. MR 454854
[11] Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
[12] Leon Simon and Neshan Wickramasekera, Stable branched minimal immersions with prescribed boundary, J. Differential Geom. 75 (2007), no. 1, 143–173. MR 2282727
[13] Neshan Wickramasekera, A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2, J. Differential Geom. 80 (2008), no. 1, 79–173. MR 2434260
[14] Neshan Wickramasekera, A general regularity theory for stable codimension 1 integral varifolds, Ann. of Math. (2) 179 (2014), no. 3, 843–1007. MR 3171756, https://doi.org/10.4007/annals.2014.179.3.2