Publications and Research
Document Type
Article
Publication Date
Summer 6-25-2017
Abstract
The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincar´e-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincar´e-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincar´e-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of q-energy from finite to infinite. At this point, a technique of p-balanced growth has been introduced to overcome difficulties of broadening from the finite q-energy in L^q spaces to an infinite q-energy in non-L^q spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincar´e-Sobolev Inequality approaching to an infinite q-energy growth.
Comments
This research work by Professor Dr. Lina Wu was supported by Faculty Development Grant in The City University of New York at Borough of Manhattan Community College.