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
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Proc. Amer. Math. Soc. 144 (2016), 1209-1221 Request permission

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.

References

William K. Allard, On the first variation of a varifold: boundary behavior, Ann. of Math. (2) 101 (1975), 418–446. MR 397520, DOI 10.2307/1970934
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, DOI 10.1007/BF01425476
Robert Finn, New estimates for equations of minimal surface type, Arch. Rational Mech. Anal. 14 (1963), 337–375. MR 157096, DOI 10.1007/BF00250712
D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377. MR 1062966, DOI 10.1007/BF01231506
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, DOI 10.1007/s00526-009-0301-y
Leobardo Rosales, Discontinuous solutions to the two-valued minimal surface equation, Adv. Calc. Var. 4 (2011), no. 4, 363–395. MR 2844510, DOI 10.1515/ACV.2011.005
Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797. MR 634285, DOI 10.1002/cpa.3160340603
Leon Simon, Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. J. 25 (1976), no. 9, 821–855. MR 412605, DOI 10.1512/iumj.1976.25.25066
Leon Simon, A Hölder estimate for quasiconformal maps between surfaces in Euclidean space, Acta Math. 139 (1977), no. 1-2, 19–51. MR 452746, DOI 10.1007/BF02392233
Leon Simon, Equations of mean curvature type in $2$ independent variables, Pacific J. Math. 69 (1977), no. 1, 245–268. MR 454854, DOI 10.2140/pjm.1977.69.245
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
Leon Simon and Neshan Wickramasekera, Stable branched minimal immersions with prescribed boundary, J. Differential Geom. 75 (2007), no. 1, 143–173. MR 2282727
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
Neshan Wickramasekera, A general regularity theory for stable codimension 1 integral varifolds, Ann. of Math. (2) 179 (2014), no. 3, 843–1007. MR 3171756, DOI 10.4007/annals.2014.179.3.2