Commenced in January 2007
Paper Count: 30843
Discovering Liouville-Type Problems for p-Energy Minimizing Maps in Closed Half-Ellipsoids by Calculus Variation Method
Abstract:The goal of this project is to investigate constant properties (called the Liouville-type Problem) for a p-stable map as a local or global minimum of a p-energy functional where the domain is a Euclidean space and the target space is a closed half-ellipsoid. The First and Second Variation Formulas for a p-energy functional has been applied in the Calculus Variation Method as computation techniques. Stokes’ Theorem, Cauchy-Schwarz Inequality, Hardy-Sobolev type Inequalities, and the Bochner Formula as estimation techniques have been used to estimate the lower bound and the upper bound of the derived p-Harmonic Stability Inequality. One challenging point in this project is to construct a family of variation maps such that the images of variation maps must be guaranteed in a closed half-ellipsoid. The other challenging point is to find a contradiction between the lower bound and the upper bound in an analysis of p-Harmonic Stability Inequality when a p-energy minimizing map is not constant. Therefore, the possibility of a non-constant p-energy minimizing map has been ruled out and the constant property for a p-energy minimizing map has been obtained. Our research finding is to explore the constant property for a p-stable map from a Euclidean space into a closed half-ellipsoid in a certain range of p. The certain range of p is determined by the dimension values of a Euclidean space (the domain) and an ellipsoid (the target space). The certain range of p is also bounded by the curvature values on an ellipsoid (that is, the ratio of the longest axis to the shortest axis). Regarding Liouville-type results for a p-stable map, our research finding on an ellipsoid is a generalization of mathematicians’ results on a sphere. Our result is also an extension of mathematicians’ Liouville-type results from a special ellipsoid with only one parameter to any ellipsoid with (n+1) parameters in the general setting.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1339760Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 592
 R. Bellman, Dynamic programming and Lagrange Multiplies, Proceedings of the National Academy of Sciences, 1956, 42(10): 767-769.
 L. F. Cheung and P. F. Leung, A remark on convex functions and p-harmonic maps,Geometriae Dedicata, 56(3), 269-270.
 F. H. Clarke Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Doctoral thesis, University of Washington, 1973. (Thesis director: R.T. Rockafellar)
 L. A. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequalities with Weights, Compositio Math., 53(1984), 259-275.
 S. Kawai, p-Harmonic maps and convex functions,Geometriae Dedicata, 74(3), 261-265.
 M. Morse, Relations between the critical points of a real function of n independent variables, Transactions of the American Mathematical Society, 1925, 27(3): 345-396.
 S. Pigola, M. Rigoli and A. G. Setti, Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds, Mathematische Zeitschrift, 258(2), 347-362.
 L. S. Pontryagin, R. V. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko The Mathematical Theory of Optimal processes, Wileylnterscience, New York, 1962.
 R. T. Rockafellar Generalized Hamiltonian Equations for Conevx problems of Lagrange, Pacific J. Math., 33:411-428, 1970.
 R. Shoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Commentarii Mathematici Helvetici 51(1), 333-341.
 S. W. Wei, The minima of the p-energy functional, Elliptic and Parabolic Methods in Geometry, A.K. Peters (1996) 171-203
 S. W. Wei, J. F. Li and L. Wu, p-Harmonic generalizations of the uniformization theorem and Bochner’s method, and geometric applications, Proceedings of the 2006 Midwest Geometry Conference, Commun. Math. Anal., Conf. 01(2008), 46-68.
 S. W. Wei and C. M. Yau, Regularity of p-energy minimizing maps and p-super-strongly unstable indices, J.Geom. Analysis, vol 4, No.2 (1994), 247-272.
 L. Wu, S. W. Wei, J. Liu and Y. Li, Discovering Geometric and Topological Properties of Ellipsoids by Curvatures, British Journal of Mathematics and Computer Science, 8(4): 318-329, 2015.