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Viscosity convex functions on Carnot groups. (English) Zbl 1057.22012
A simply connected (nilpotent) Lie group \(G\) with Lie algebra \(\mathfrak{g}\) decomposed as \(\mathfrak{g} = V_1 + V_2 + \cdots + V_r\) with \([V_1, V_j] = V_{j+1}\) for \(j < r\) and \([V_1,V_r] = 0\) is called a Carnot group. Given a domain \(\Omega\subset G\), a function \(u:\Omega\to\mathbb{R}\) is said to be \(h\)-convex whenever \(u| [p, q]\) is convex for any line segment \([p, q] = \exp ([0, 1]\cdot X)\subset\Omega\) with \(X\in V_1\). If \(u:\Omega\to\mathbb{R}\) is upper semicontinuous, then \(u\) is said to be \(v\)-convex whenever its horizontal hessian \(\nabla ^2_h u = \nabla ^2u| V_1\times V_1\) is nonnegative in the sense of viscosity; that is \(\sum _{i,j=1}^m \xi_i\xi_jX_iX_j\phi (p)\neq 0\) for any \(p\in\Omega\), \(\xi_1, \dots , \xi_m\in\mathbb{R}\) (\(m = \dim V_1\)), a basis \(X_1, \dots X_m\) of \(V_1\), and any C\(^2\)-function \(\phi\) “touching” \(u\) at \(p\) from above. Theorem. Any upper semicontinuous \(v\)-convex function on a Carnot group is \(h\)-convex.

MSC:
22E30 Analysis on real and complex Lie groups
22E25 Nilpotent and solvable Lie groups
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[1] Thomas Bieske, On \infty -harmonic functions on the Heisenberg group, Comm. Partial Differential Equations 27 (2002), no. 3-4, 727 – 761. · Zbl 1090.35063 · doi:10.1081/PDE-120002872 · doi.org
[2] Zoltán M. Balogh and Matthieu Rickly, Regularity of convex functions on Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 847 – 868. · Zbl 1121.43007
[3] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. · Zbl 0834.35002
[4] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1 – 67. · Zbl 0755.35015
[5] Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu, Notions of convexity in Carnot groups, Comm. Anal. Geom. 11 (2003), no. 2, 263 – 341. · Zbl 1077.22007 · doi:10.4310/CAG.2003.v11.n2.a5 · doi.org
[6] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[7] G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. · Zbl 0508.42025
[8] R. Jensen, P.-L. Lions, and P. E. Souganidis, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc. 102 (1988), no. 4, 975 – 978. · Zbl 0662.35048
[9] P. Juutinen, G. Lu, J. Manfredi, B. Stroffolini. In preparation.
[10] Guozhen Lu, Juan J. Manfredi, and Bianca Stroffolini, Convex functions on the Heisenberg group, Calc. Var. Partial Differential Equations 19 (2004), no. 1, 1 – 22. · Zbl 1072.49019 · doi:10.1007/s00526-003-0190-4 · doi.org
[11] C. Y. Wang, The Aronsson equation for absolute minimizers of \(L^{\infty}\)-functionals associated with vector fields satisfying Hörmander’s condition. To appear in Trans. Amer. Math. Soc. · Zbl 1192.35038
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