August  2022, 15(8): 2209-2214. doi: 10.3934/dcdss.2022032

Some rigidity results for minimal graphs over unbounded Euclidean domains

LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France

*Corresponding author: Alberto Farina

Al caro amico Maurizio con grande affetto e grande stima
The author thanks L. Mari for a careful reading of a first version of this article as well as for interesting discussions on the subject

Received  December 2021 Published  August 2022 Early access  February 2022

We prove some new rigidity results for minimal graphs over unbounded Euclidean domains. In particular we prove that a positive minimal graph over an open affine half-space, and under the homogenous Dirichlet boundary condition, must be an affine function.

Citation: Alberto Farina. Some rigidity results for minimal graphs over unbounded Euclidean domains. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2209-2214. doi: 10.3934/dcdss.2022032
References:
[1]

F. J. Jr. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math., 84 (1966), 277-292.  doi: 10.2307/1970520.

[2]

S. Bernstein, Über ein geometrisches theorem und seine anwendung auf die partiellen differentialgleichungen vom elliptischen typus, Math. Z., 26 (1927), 551-558.  doi: 10.1007/BF01475472.

[3]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[4]

E. BombieriE. De Giorgi and M. Miranda, Una maggiorazione a priori relativa alle ipersuperficie minimali non parametriche, Arch. Rat. Mech. Anal., 32 (1969), 255-267.  doi: 10.1007/BF00281503.

[5]

E. Bombieri and E. Giusti, Harnack's inequality for elliptic differential equations on minimal surfaces, Invent. Math., 15 (1972), 24-46.  doi: 10.1007/BF01418640.

[6]

E. D. Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85. 

[7]

A. Farina, A Bernstein-type result for the minimal surface equation, Ann. Scuola Norm. Sup. Pisa, 14 (2015), 1231-1237. 

[8]

A. Farina, A sharp Bernstein-type theorem for entire minimal graphs, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 123, 5 pp. doi: 10.1007/s00526-018-1392-0.

[9]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.

[10]

E. GonzalezU. Massari and M. Miranda, On minimal cones, Appl. Anal., 65 (1997), 135-143.  doi: 10.1080/00036819708840554.

[11]

E. Hopf, On S. Bernstein's theorem on surfaces $z(x, y)$ of nonpositive curvature, Proc. Amer. Math. Soc., 1 (1950), 80-85.  doi: 10.2307/2032438.

[12]

U. Massari and M. Miranda, Minimal Surfaces of Codimension One, Notas de Matemática, North Holland, Amsterdam, 1984.

[13]

M. Miranda, Un principio di massimo forte per le frontiere minimali, Rend. Sem. Mat. Univ. Padova, 45 (1971), 355-366. 

[14]

M. Miranda, Superfici minime illimitate, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 313-322. 

[15]

M. P. Moschen, Principio di massimo forte per le frontiere di misura minima, Ann. Univ. Ferrara Sez. VII (N.S.), 23 (1977), 165-168.  doi: 10.1007/BF02825995.

[16]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.

[17]

J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math., 88 (1968), 62-105.  doi: 10.2307/1970556.

show all references

References:
[1]

F. J. Jr. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math., 84 (1966), 277-292.  doi: 10.2307/1970520.

[2]

S. Bernstein, Über ein geometrisches theorem und seine anwendung auf die partiellen differentialgleichungen vom elliptischen typus, Math. Z., 26 (1927), 551-558.  doi: 10.1007/BF01475472.

[3]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[4]

E. BombieriE. De Giorgi and M. Miranda, Una maggiorazione a priori relativa alle ipersuperficie minimali non parametriche, Arch. Rat. Mech. Anal., 32 (1969), 255-267.  doi: 10.1007/BF00281503.

[5]

E. Bombieri and E. Giusti, Harnack's inequality for elliptic differential equations on minimal surfaces, Invent. Math., 15 (1972), 24-46.  doi: 10.1007/BF01418640.

[6]

E. D. Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85. 

[7]

A. Farina, A Bernstein-type result for the minimal surface equation, Ann. Scuola Norm. Sup. Pisa, 14 (2015), 1231-1237. 

[8]

A. Farina, A sharp Bernstein-type theorem for entire minimal graphs, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 123, 5 pp. doi: 10.1007/s00526-018-1392-0.

[9]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.

[10]

E. GonzalezU. Massari and M. Miranda, On minimal cones, Appl. Anal., 65 (1997), 135-143.  doi: 10.1080/00036819708840554.

[11]

E. Hopf, On S. Bernstein's theorem on surfaces $z(x, y)$ of nonpositive curvature, Proc. Amer. Math. Soc., 1 (1950), 80-85.  doi: 10.2307/2032438.

[12]

U. Massari and M. Miranda, Minimal Surfaces of Codimension One, Notas de Matemática, North Holland, Amsterdam, 1984.

[13]

M. Miranda, Un principio di massimo forte per le frontiere minimali, Rend. Sem. Mat. Univ. Padova, 45 (1971), 355-366. 

[14]

M. Miranda, Superfici minime illimitate, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 313-322. 

[15]

M. P. Moschen, Principio di massimo forte per le frontiere di misura minima, Ann. Univ. Ferrara Sez. VII (N.S.), 23 (1977), 165-168.  doi: 10.1007/BF02825995.

[16]

J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.  doi: 10.1002/cpa.3160140329.

[17]

J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math., 88 (1968), 62-105.  doi: 10.2307/1970556.

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