Applying p-Balanced Energy Technique to Solve Liouville-Type Problems in Calculus
We are interested in solving Liouville-type problems to explore constancy properties for maps or differential forms on Riemannian manifolds. Geometric structures on manifolds, the existence of constancy properties for maps or differential forms, and energy growth for maps or differential forms are intertwined. In this article, we concentrate on discovery of solutions to Liouville-type problems where manifolds are Euclidean spaces (i.e. flat Riemannian manifolds) and maps become real-valued functions. Liouville-type results of vanishing properties for functions are obtained. The original work in our research findings is to extend the q-energy for a function from finite in Lq space to infinite in non-Lq space by applying p-balanced technique where q = p = 2. Calculation skills such as Hölder's Inequality and Tests for Series have been used to evaluate limits and integrations for function energy. Calculation ideas and computational techniques for solving Liouville-type problems shown in this article, which are utilized in Euclidean spaces, can be universalized as a successful algorithm, which works for both maps and differential forms on Riemannian manifolds. This innovative algorithm has a far-reaching impact on research work of solving Liouville-type problems in the general settings involved with infinite energy. The p-balanced technique in this algorithm provides a clue to success on the road of q-energy extension from finite to infinite.
 R. Schoen, and S. T. Yau, “Harmonic maps and topology of stable hypersurfaces and manifolds with non-negative Ricci curvature,” Comment. Math. Helv., vol. 51, no. 1, pp. 333-341, 1976.
 R. Greene, and H. Wu, “Harmonic forms on non-compact Riemannian and Kahler manifolds,” The Michigan Mathematical Journal, vol. 28, no. 1, pp. 63-81, 1981.
 S. Kawai, “P-Harmonic maps and convex functions,” Geometriae Dedicata, vol. 74, no. 3, pp. 261-265, 1994.
 L.-F. Cheung, and P.-F. Leung, “A remark on convex functions and p-harmonic maps,” Geometriae Dedicata, vol. 56, no. 3, pp. 269-270, 1995.
 X. Zhang, “A note on p-harmonic 1-form on complete manifolds,” Canad. Math. Bull., vol. 44, no. 3, pp. 376-384, 2001.
 S. Pigola, M. Rigoli, and A. G. Setti, “Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds,” Mathematische Zeitschrift, vol. 258, no. 2, pp. 34-362, 2008.
 H. Wu, Elementary introduction of Riemannian manifolds (Chinese version), Peking University Press, Beijing, China, 1989.
 S. W. Wei, J. Li, and L. Wu, “Generalizations of the uniformization theorem and Bochner’s method in p-harmonic geometry,” Commun. Math. Anal., Conf., vol. 1, pp. 46-68, 2008.
 S. W. Wei, and L. Wu, “Vanishing theorems for 2-balanced harmonic forms,” Global Journal of Pure and Applied Mathematics, vol. 11, no. 2, pp. 745-753, 2015.
 L. Wu, “Solving Liouville-type problems on manifolds with Poincare-Sobolev Inequality by broadening q-energy from finite to infinite,” Journal of Mathematics Research, vol. 9, no. 4, pp. 1-10, 2017.
 L. Wu, and Y. Li, “Equivalence between a harmonic form and a closed co-closed form in both Lq and non-Lq spaces,” European Journal of Mathematical Sciences, vol. 3, no. 1, pp. 1-13, 2017.
 L. Wu, and Y. Li, “Generalizing Liouville-type problems for differential 1-forms from Lq spaces to non-Lq spaces,” International Journal of Mathematical Analysis, vol. 10, no. 28, pp. 1375-1387, 2016.
 L. Wu, J. Liu, and Y. Li, “Discovering Liouville-type problems for p-energy minimizing maps in closed half-ellipsoids by Calculus variation method,” International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, vol. 10, no. 10, pp. 496-502, 2016.