- Research Article
- Open Access
© En-Bing Lin 2010
- Received: 31 December 2009
- Accepted: 7 February 2010
- Published: 4 March 2010
We prove a characterization of a nonhomogeneous A-harmonic equation and describe its generalization. We also point out its connection with 1-Harmonic equation.
- Continuous Function
- Weak Solution
- Open Problem
- Usual Sense
- Harmonic Extension
Both A-harmonic equations and -harmonic geometry are rich subjects [1–5]. Many results on both topics have been derived, respectively, but there are very few papers relating both subjects. In this paper, we will connect these two subjects by extending several results from 1-Harmonic functions to A-Harmonic functions.
In this paper, we characterize subsolutions of (1.1) and indicate its generalization to (1.2). We first recall some results in homogeneous A-harmonic equations in the following section, followed by the main results in Section 3. Some open problems are discussed in the last section.
It was shown in  that very weak solutions of (1.2) in fact weak solutions of (1.2) in the usual sense.
an -harmonic tensor in of satisfies the -harmonic equation (2.1) in .
Consider the space of differential -forms being an A-harmonic tensor in a domain , and Assume that , and for some . Then the following local weighted Poincare inequality for A-harmonic tensors was proved in . There exists a constant independent of such that
In what follows, we prove an A-harmonic analog of 1-harmonic equations.
Let M be a complete noncompact Riemannian. For any and any pair of positive numbers with , there exist a rotational symmetric Lipschitz continuous function and a constant (independent of ) with the following properties:
Let be a weak subsolution of -harmonic equation (1.1) with constant 1-tension field , that is, in the distribution sense. Then is an A-harmonic function.
It would be interesting to find similar results of Section 2 for nonhomogeneous -harmonic equations. It would also be interesting to seek analogs of 1-harmonic applications in calibration geometry. The extension of 1-harmonic functions to -harmonic functions on hyperbolic spaces and their associated spaces could be explored.
To conclude this paper, we state another -harmonic extension of 1-harmonic result , that is an immediate consequence of Corollary 3.3.
The author wishes to express sincere gratitude to Professor S. Walter Wei for many helpful suggestions and encouragements without which this would have not been written.
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