\r\nproperties (called the Liouville-type Problem) for a p-stable map

\r\nas a local or global minimum of a p-energy functional where

\r\nthe domain is a Euclidean space and the target space is a

\r\nclosed half-ellipsoid. The First and Second Variation Formulas

\r\nfor a p-energy functional has been applied in the Calculus

\r\nVariation Method as computation techniques. Stokes’ Theorem,

\r\nCauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and

\r\nthe Bochner Formula as estimation techniques have been used to

\r\nestimate the lower bound and the upper bound of the derived

\r\np-Harmonic Stability Inequality. One challenging point in this project

\r\nis to construct a family of variation maps such that the images

\r\nof variation maps must be guaranteed in a closed half-ellipsoid.

\r\nThe other challenging point is to find a contradiction between the

\r\nlower bound and the upper bound in an analysis of p-Harmonic

\r\nStability Inequality when a p-energy minimizing map is not constant.

\r\nTherefore, the possibility of a non-constant p-energy minimizing

\r\nmap has been ruled out and the constant property for a p-energy

\r\nminimizing map has been obtained. Our research finding is to explore

\r\nthe constant property for a p-stable map from a Euclidean space into

\r\na closed half-ellipsoid in a certain range of p. The certain range of

\r\np is determined by the dimension values of a Euclidean space (the

\r\ndomain) and an ellipsoid (the target space). The certain range of p

\r\nis also bounded by the curvature values on an ellipsoid (that is, the

\r\nratio of the longest axis to the shortest axis). Regarding Liouville-type

\r\nresults for a p-stable map, our research finding on an ellipsoid is a

\r\ngeneralization of mathematicians’ results on a sphere. Our result is

\r\nalso an extension of mathematicians’ Liouville-type results from a

\r\nspecial ellipsoid with only one parameter to any ellipsoid with (n+1)

\r\nparameters in the general setting.","references":"[1] R. Bellman, Dynamic programming and Lagrange Multiplies,\r\nProceedings of the National Academy of Sciences, 1956, 42(10):\r\n767-769.\r\n[2] L. F. Cheung and P. F. Leung, A remark on convex functions and\r\np-harmonic maps,Geometriae Dedicata, 56(3), 269-270.\r\n[3] F. H. Clarke Necessary Conditions for Nonsmooth Problems in Optimal\r\nControl and the Calculus of Variations, Doctoral thesis, University of\r\nWashington, 1973. (Thesis director: R.T. Rockafellar)\r\n[4] L. A. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation\r\nInequalities with Weights, Compositio Math., 53(1984), 259-275.\r\n[5] S. Kawai, p-Harmonic maps and convex functions,Geometriae Dedicata,\r\n74(3), 261-265.\r\n[6] M. Morse, Relations between the critical points of a real function of\r\nn independent variables, Transactions of the American Mathematical\r\nSociety, 1925, 27(3): 345-396.\r\n[7] S. Pigola, M. Rigoli and A. G. Setti, Constancy of p-harmonic maps\r\nof finite q-energy into non-positively curved manifolds, Mathematische\r\nZeitschrift, 258(2), 347-362.\r\n[8] L. S. Pontryagin, R. V. Boltyanskii, R. V. Gamkrelidze, and\r\nE. F. Mischenko The Mathematical Theory of Optimal processes,\r\nWileylnterscience, New York, 1962.\r\n[9] R. T. Rockafellar Generalized Hamiltonian Equations for Conevx\r\nproblems of Lagrange, Pacific J. Math., 33:411-428, 1970.\r\n[10] R. Shoen and S. T. Yau, Harmonic maps and the topology of\r\nstable hypersurfaces and manifolds with non-negative Ricci curvature,\r\nCommentarii Mathematici Helvetici 51(1), 333-341.\r\n[11] S. W. Wei, The minima of the p-energy functional, Elliptic and Parabolic\r\nMethods in Geometry, A.K. Peters (1996) 171-203\r\n[12] S. W. Wei, J. F. Li and L. Wu, p-Harmonic generalizations of\r\nthe uniformization theorem and Bochner\u2019s method, and geometric\r\napplications, Proceedings of the 2006 Midwest Geometry Conference,\r\nCommun. Math. Anal., Conf. 01(2008), 46-68.\r\n[13] S. W. Wei and C. M. Yau, Regularity of p-energy minimizing maps and\r\np-super-strongly unstable indices, J.Geom. Analysis, vol 4, No.2 (1994),\r\n247-272.\r\n[14] L. Wu, S. W. Wei, J. Liu and Y. Li, Discovering Geometric and\r\nTopological Properties of Ellipsoids by Curvatures, British Journal of\r\nMathematics and Computer Science, 8(4): 318-329, 2015.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 118, 2016"}