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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 |
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.
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. Bombieri, E. De Giorgi and E. Giusti,
Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.
doi: 10.1007/BF01404309. |
| [4] |
E. Bombieri, E. 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. Gonzalez, U. 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. Bombieri, E. De Giorgi and E. Giusti,
Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-268.
doi: 10.1007/BF01404309. |
| [4] |
E. Bombieri, E. 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. Gonzalez, U. 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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