Examples of holomorphic functions vanishing to infinite order at the boundary

Hirsch, Jonas

doi:10.1090/tran/7192

Examples of holomorphic functions vanishing to infinite order at the boundary
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Trans. Amer. Math. Soc. 370 (2018), 4249-4271 Request permission

Abstract:

We present examples of holomorphic functions that vanish to infinite order at points at the boundary of their domain of definition. They give rise to examples of Dirichlet minimizing $Q$-valued functions indicating that “higher”-regularity boundary results are difficult. Furthermore we discuss some implication to branching and vanishing phenomena in the context of minimal surfaces, $Q$-valued functions, and unique continuation.

References

Frederick J. Almgren Jr., Almgren’s big regularity paper, World Scientific Monograph Series in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. $Q$-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2; With a preface by Jean E. Taylor and Vladimir Scheffer. MR 1777737
Jeff Cheeger, Aaron Naber, and Daniele Valtorta, Critical sets of elliptic equations, Comm. Pure Appl. Math. 68 (2015), no. 2, 173–209. MR 3298662, DOI 10.1002/cpa.21518
Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
Camillo De Lellis, Carlo Romano Grisanti, and Paolo Tilli, Regular selections for multiple-valued functions, Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 79–95. MR 2044333, DOI 10.1007/s10231-003-0081-5
Camillo De Lellis and Emanuele Nunzio Spadaro, $Q$-valued functions revisited, Mem. Amer. Math. Soc. 211 (2011), no. 991, vi+79. MR 2663735, DOI 10.1090/S0065-9266-10-00607-1
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J. 35 (1986), no. 2, 245–268. MR 833393, DOI 10.1512/iumj.1986.35.35015
Nicola Garofalo and Fang-Hua Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), no. 3, 347–366. MR 882069, DOI 10.1002/cpa.3160400305
Robert Gulliver, A minimal surface with an atypical boundary branch point, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 211–228. MR 1173043
Robert Gulliver and Frank David Lesley, On boundary branch points of minimizing surfaces, Arch. Rational Mech. Anal. 52 (1973), 20–25. MR 346641, DOI 10.1007/BF00249089
Jonas Hirsch, Boundary regularity of Dirichlet minimizing $Q$-valued functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1353–1407. MR 3616337
B. Krummel and N. Wickramasekera, Fine properties of branch point singularities: Two-valued harmonic functions, arXiv e-prints (2013).
Aaron Naber and Daniele Valtorta, Volume estimates on the critical sets of solutions to elliptic PDEs, Comm. Pure Appl. Math. 70 (2017), no. 10, 1835–1897. MR 3688031, DOI 10.1002/cpa.21708
Robert Osserman, A proof of the regularity everywhere of the classical solution to Plateau’s problem, Ann. of Math. (2) 91 (1970), 550–569. MR 266070, DOI 10.2307/1970637
Leon Simon and Neshan Wickramasekera, A frequency function and singular set bounds for branched minimal immersions, Comm. Pure Appl. Math. 69 (2016), no. 7, 1213–1258. MR 3503021, DOI 10.1002/cpa.21642
Emanuele Nunzio Spadaro, Complex varieties and higher integrability of Dir-minimizing $Q$-valued functions, Manuscripta Math. 132 (2010), no. 3-4, 415–429. MR 2652440, DOI 10.1007/s00229-010-0353-5