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Existence and non-existence of minimal graphic and \(p\)-harmonic functions. (English) Zbl 1445.53031

The paper under review studies the minimal graph equation: \(\operatorname{div}\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}=0\) and \(p\)-harmonic equation: \(\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0\).
A main result of the paper shows that any solution of the minimal graph equation that is bounded from below and has at most linear growth on a complete Riemannian manifold with asymptotically nonnegative sectional curvature and only one end is constant. Here, a Riemannian manifold is said to have asymptotically nonnegative sectional curvature means that \(\mathrm{K}_M(p_x)\geq -\lambda(d(o,x))\) for a certain continuous decreasing function \(\lambda: [0,\infty)\to [0,\infty)\).
The key in the proof of the main theorem is to obtain a uniform gradient estimate for the solutions of the minimal graph equation since the previous gradient estimate obtained in [J. Spruck, Pure Appl. Math. Q. 3, No. 3, 785–800 (2007; Zbl 1145.53048)] and [H. Rosenberg et al., J. Differ. Geom. 95, No. 2, 321–336 (2013; Zbl 1291.53075)] under the assumption on the Ricci curvature is not applicable here.
The paper also proves the existence of solutions to the minimal graph and \(p\)-harmonic equations on rotationally symmetric \(n\)-dimensional Cartan-Hadamard manifolds under certain assumptions on the radial sectional curvatures.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
31C45 Other generalizations (nonlinear potential theory, etc.)
58J32 Boundary value problems on manifolds
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[1] Abresch, U.. Lower curvature bounds, Toponogov’s theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4)18 (1985), 651-670. · Zbl 0595.53043
[2] Bombieri, E., De Giorgi, E. and Miranda, M.. Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Rational Mech. Anal.32 (1969), 255-267. · Zbl 0184.32803
[3] Buser, P.. A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4)15 (1982), 213-230. · Zbl 0501.53030
[4] Cai, M.. Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set. Bull. Amer. Math. Soc. (N.S.)24 (1991), 371-377. · Zbl 0728.53026
[5] Casteras, J.-B., Heinonen, E. and Holopainen, I.. Solvability of minimal graph equation under pointwise pinching condition for sectional curvatures. J. Geom. Anal.27 (2017a), 1106-1130. · Zbl 1372.58015
[6] Casteras, J.-B., Holopainen, I. and Ripoll, J. B.. On the Asymptotic Dirichlet Problem for the Minimal Hypersurface Equation in a Hadamard Manifold. Potential Anal.47 (2017b), 485-501. · Zbl 1380.58020
[7] Casteras, J.-B., Heinonen, E. and Holopainen, I.. Dirichlet problem for f-minimal graphs. J. Anal. Math. To appear. · Zbl 1427.53079
[8] Casteras, J.-B., Holopainen, I. and Ripoll, J.B.. Asymptotic Dirichlet problem for \({\cal A}\)-harmonic and minimal graph equations in a Cartan-Hadamard manifolds. Comm. Anal. Geom. To appear. · Zbl 1430.53047
[9] Dajczer, M. and De Lira, J. H. S.. Entire bounded constant mean curvature Killing graphs. J. Math. Pures Appl. (9)103 (2015), 219-227. · Zbl 1333.53087
[10] Dajczer, M. and De Lira, J. H. S.. Entire unbounded constant mean curvature Killing graphs. Bull. Braz. Math. Soc. (NS)48 (2017), 187-198. · Zbl 1383.53047
[11] Ding, Q., Jost, J. and Xin, Y.. Minimal graphic functions on manifolds of nonnegative Ricci curvature. Comm. Pure Appl. Math.69 (2016), 323-371. · Zbl 1335.53044
[12] Eberlein, P. and O’Neill, B.. Visibility manifolds. Pacific J. Math.46 (1973), 45-109. · Zbl 0264.53026
[13] Fornari, S. and Ripoll, J.. Killing fields, mean curvature, translation maps. Illinois J. Math.48 (2004), 1385-1403. · Zbl 1072.53020
[14] Greene, R. E. and Wu, H.. Function theory on manifolds which possess a pole, volume 699 of Lecture Notes in Mathematics (Berlin: Springer, 1979). · Zbl 0414.53043
[15] Greene, R. E. and Wu, H.. Gap theorems for noncompact Riemannian manifolds. Duke Math. J.49 (1982), 731-756. · Zbl 0513.53045
[16] Grigor’yan, A. and Saloff-Coste, L.. Stability results for Harnack inequalities. Ann. Inst. Fourier (Grenoble)55 (2005), 825-890. · Zbl 1115.58024
[17] Gromov, M.. Metric structures for Riemannian and non-Riemannian spaces. In Modern Birkhäuser Classics, english edn (ed. Katz, M., Pansu, P. and Semmes, S.). Based on the 1981 French original, With appendices, Translated from the French by Sean Michael Bates (Boston, MA: Birkhäuser Boston, Inc., 2007).
[18] Heinonen, E.. Asymptotic Dirichlet problem for \({\cal A}\)-harmonic functions on manifolds with pinched curvature. Potential Anal.46 (2017), 63-74. · Zbl 1360.58019
[19] Heinonen, J., Kilpeläinen, T. and Martio, O.. Nonlinear potential theory of degenerate elliptic equations (New York: The Clarendon Press, Oxford University Press, 1993). Oxford Science Publications. · Zbl 0780.31001
[20] Holopainen, I.. Volume growth, Green’s functions, and parabolicity of ends. Duke Math. J.97 (1999), 319-346. · Zbl 0955.31003
[21] Holopainen, I.. Asymptotic Dirichlet problem for the p-Laplacian on Cartan-Hadamard manifolds. Proc. Amer. Math. Soc.130 (2002), 3393-3400, (electronic). · Zbl 1016.58021
[22] Holopainen, I. and Ripoll, J. B.. Nonsolvability of the asymptotic Dirichlet problem for some quasilinear elliptic PDEs on Hadamard manifolds. Rev. Mat. Iberoam.31 (2015), 1107-1129. · Zbl 1330.58017
[23] Holopainen, I. and Vähäkangas, A.. Asymptotic Dirichlet problem on negatively curved spaces. J. Anal.15 (2007), 63-110. · Zbl 1202.58017
[24] Kasue, A.. A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. École Norm. Sup. (4)21 (1988), 593-622. · Zbl 0662.53032
[25] Li, P. and Tam, L.-F.. Green’s functions, harmonic functions, and volume comparison. J. Differential Geom.41 (1995), 277-318. · Zbl 0827.53033
[26] Liu, Z.-D.. Ball covering property and nonnegative Ricci curvature outside a compact set. In Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pp. 459-464 (ed. Greene, R. and Yau, S.-T.) (Providence, RI: Amer. Math. Soc., 1993). · Zbl 0788.53028
[27] March, P.. Brownian motion and harmonic functions on rotationally symmetric manifolds. Ann. Probab.14 (1986), 793-801. · Zbl 0593.60078
[28] Ripoll, J. and Telichevesky, M.. Complete minimal graphs with prescribed asymptotic boundary on rotationally symmetric Hadamard surfaces. Geom. Dedicata161 (2012), 277-283. · Zbl 1254.53014
[29] Rosenberg, H., Schulze, F. and Spruck, J.. The half-space property and entire positive minimal graphs in M × ℝ. J. Differential Geom.95 (2013), 321-336. · Zbl 1291.53075
[30] Spruck, J.. Interior gradient estimates and existence theorems for constant mean curvature graphs in M^n × R. Pure Appl. Math. Q.3 (2007), 785-800. · Zbl 1145.53048
[31] Vähäkangas, A.. Bounded p-harmonic functions on models and Cartan-Hadamard manifolds. Unpublished licentiate thesis, Department of Mathematics and Statistics, University of Helsinki, 2006.
[32] Vähäkangas, A.. Dirichlet problem at infinity for \({\cal A}\)-harmonic functions. Potential Anal.27 (2007), 27-44. · Zbl 1182.31014
[33] Vähäkangas, A.. Dirichlet problem on unbounded domains and at infinity. Reports in Mathematics, Preprint 499, Department of Mathematics and Statistics, University of Helsinki, 2009.
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