Abstract
We show that there are no entire, positive, stable solutions in ℝn of the Euler equation corresponding to the singular variational integral\(\smallint u^\alpha \sqrt {1 + |Du|^2 } dx\),α>0, ifα+n<5.236.... Furthermore we prove a related result for smooth boundaries of leastα-energy ∫|x n+1|α|D ϕU | in ℝn+1.
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Almgren, F.J.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem. Ann. Math84, 277–292 (1966)
Bemelmans, J., Dierkes, U.: On a singular variational integral with linear growth. Arch. Ration. Mech. Anal.100, 83–103 (1987)
Bernstein, S.: Sur un théoreme de géometrie et ses applications aux équations aux dérivées partielles du type elliptique. Commun. Soc. Math. Kharkov (2)15, 38–45 (1915–1917)
Böhme, R., Hildebrandt, S., Tausch, E.: The two-dimensional analogue of the catenary. Pac. J. Math.88, 247–278 (1980)
Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math.7 243–268 (1969)
Bombieri, E., De Giorgi, E., Miranda, M.: Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Ration. Mech. Anal.32, 255–267 (1969)
Caffarelli, L., Nirenberg, L., Spruck, J.: On a form of Bernstein's Theorem. (Preprint)
Chern, S.S., Osserman, R.: Complete minimal surfaces in euclideann-space. J. Anal. Math.19, 15–34 (1967)
De Giorgi, E.: Una estensione del teorema di Bernstein. Ann. Sc. Norm. Super. Pisa, III19, 79–85 (1965)
Dierkes, U.: Minimal hypercones andC 0,1/2 minimizers for a singular variational problem. Indiana Univ. Math. J.37 (4), 841–863 (1988)
Dierkes, U.: On the non-existence of energy stable minimal cones. Ann. Inst. Henri Poincaré, Anal. Non Linéaire,7 (6), 589–601 (1990)
Dierkes, U.: über singuläre Lösungen gewisser mehrdimensionaler Variationsprobleme. Ann. Univ. Sarav.3 (2), 38–108 (1990)
Dierkes, U.: On singular variational problems. Proc. Cent. Math. Appl., Aust. Natl. Univ. Canberra26, 89–106 (1991)
Dierkes, U., Huisken, G.: The N-dimensional analogue of the catenary: existence and non-existence. Pac. J. Math141 (1) 47–54 (1990)
Ecker, K., Huisken, G.: A Bernstein result for minimal graphs of controlled growth. J. Differ. Geom.31, 397–400 (1990)
Fischer-Colbrie, D.: Some rigidity theorems for minimal submanifolds of the sphere. Acta Math.145, 29–46 (1980)
Fleming, W.H.: On the oriented Plateau problem. Rend. Circ. Mat. Palermo2, 1–22 (1962)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. (Grundlehren math. Wiss., vol 224) Berlin Heidelberg New York: Springer 1977, 2nd edn. 1983
Heinz, E.: über die Lösungen der Minimalflächengleichung. Nachr. Akad. Wiss. Gött. Math. Phys. Kl. II 1952: 51–56
Hildebrandt, S., Jost, J., Widman, K.O.: Harmonic mappings and minimal submanifolds. Invent. Math62, 269–298 (1980)
Jenkins, H.: On two-dimensional variational problems in parametric form. Arch. Ration. Mech. Anal.8, 181–206 (1961)
Keiper, J.B.: The axially symmetricn-tectum. (Preprint)
Lagrange, J.L.: Mécanique analytique quatrième édition. ∄uvres tome onzième
Moser, J.: On Harnack's theorem for elliptic differential equations. Commun. Pure Appl. Math14, 557–591 (1961)
Nitsche, J.C.C.: Lectures on minimal surfaces. vol. 1. Cambridge: Cambridge University Press 1989
Nitsche, J.C.C.: A nonuniqueness theorem for the two-dimensional analogue of the catenary. Analysis6, 143–156 (1986)
Otto, F.: Zugbeanspruchte Konstruktionen. Bd. I, II. Berlin Frankfurt/M. Wien: Ullstein 1962, 1966
Poisson: Sur les surfaces élastique. Mem. Cl. Sci. Math. Phys. Inst. France 1812, deux. p. 167–225
Schoen, R., Simon, L., Yau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math.134, 275–288 (1975)
Simon, L.: Remarks on curvature estimates for minimal hyperfaces. Duke Math. J.43 (3), 545–553 (1976)
Simon, L.: On some extensions of Bernstein's theorem. Math. Z.154, 265–273 (1977)
Simon, L.: Entire solutions of the minimal surface equation. Preprint, CMA -R25-87 Austral. Nat. Univ. Canberra 1987
Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math.88, 62–105 (1968)