Journal of Inequalities and Applications
VolumeΒ 2010Β (2010), Article IDΒ 346308, 4 pages
doi:10.1155/2010/346308
Research Article

On Nonhomogeneous 𝐴 -Harmonic Equations and 1-Harmonic Equations

En-Bing Lin

Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

Received 31 December 2009; Accepted 7 February 2010

Academic Editor: ShusenΒ Ding

Copyright Β© 2010 En-Bing Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove a characterization of a nonhomogeneous A-harmonic equation and describe its generalization. We also point out its connection with 1-Harmonic equation.

1. Introduction

Both A-harmonic equations and 𝑝 -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.

We consider the following setting: a 𝐢 1 function 𝑓 ∢ 𝑅 𝑛 β†’ 𝑅 is said to be A-harmonic if it is a weak solution of A-harmonic equation

ξ‚΅ d i v 𝐴 π‘₯ , βˆ‡ 𝑓 | | | | ξ‚Ά βˆ‡ 𝑓 = 0 , ( 1 . 1 ) where | βˆ‡ 𝑓 | is the length of the gradient βˆ‡ 𝑓 of 𝑓 , and for a 𝐢 2 function 𝑓 without a critical point, d i v ( βˆ‡ 𝑓 / | βˆ‡ 𝑓 | ) is said to be the 1-tension field of 𝑓 .

Let Ξ© be an open subset of 𝑅 𝑛 , 𝑛 β‰₯ 2 . Consider the following second-order divergence-type elliptic equation:

d i v 𝐴 ( π‘₯ , βˆ‡ 𝑓 ( π‘₯ ) ) = 0 , ( 1 . 2 ) where 𝐴 ∢ Ξ© x 𝑅 𝑛 β†’ 𝑅 𝑛 satisfies

(i) | 𝐴 ( π‘₯ , 𝑒 ) | ≀ πœ† 𝑀 ( π‘₯ ) | 𝑒 | 𝑝 βˆ’ 1 , (ii) ⟨ 𝐴 ( π‘₯ , 𝑒 ) , 𝑒 ⟩ β‰₯ πœ† βˆ’ 1 𝑀 ( π‘₯ ) | 𝑒 | 𝑝 ,

where 1 < 𝑝 < ∞ , πœ† β‰₯ 1 are two fixed constants, and 𝑀 ( π‘₯ ) is called a weight if 𝑀 ∈ 𝐿 1 l o c ( 𝑅 𝑛 ) and 𝑀 > 0 a.e. Also, in general 𝑑 πœ‡ = 𝑀 𝑑 π‘₯ where 𝑀 is a weight.

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.

2. Homogeneous 𝐴 -Harmonic Equations

Definition. A function 𝑓 ∈ π‘Š 1 , π‘Ÿ l o c ( Ξ© , 𝑀 ) , m a x { 1 , 𝑝 βˆ’ 1 } ≀ π‘Ÿ < 𝑝 is called a very weak solution of (1.2) if ξ€œ Ξ© < 𝐴 ( π‘₯ , βˆ‡ 𝑓 ) , βˆ‡ πœ™ > 𝑑 π‘₯ = 0 ( 2 . 1 ) for all πœ™ ∈ π‘Š 1 , ( π‘Ÿ / ( π‘Ÿ βˆ’ 𝑝 + 1 ) ) ( Ξ© , 𝑀 ) with compact support.

It was shown in [4] that very weak solutions of (1.2) in fact weak solutions of (1.2) in the usual sense.

Definition 2.2. 𝑒 an 𝐴 -harmonic tensor in Ξ© of 𝑒 satisfies the 𝐴 -harmonic equation (2.1) in Ξ© .

Consider the space of differential 𝑙 -forms 𝐷 β€² ( Ξ© , Ξ› 𝑙 ) , 𝑒 ∈ 𝐷 β€² ( Ξ© , Ξ› 𝑙 ) being an A-harmonic tensor in a domain Ξ© βŠ‚ 𝑅 𝑛 , and 𝑑 𝑒 ∈ 𝐿 𝑠 l o c ( Ξ© , Ξ› 𝑙 + 1 ) , 𝑙 = 0 , 1 , … , 𝑛 . Assume that 𝜎 > 1 , 1 < 𝑠 < ∞ , and 𝑀 ∈ 𝐴 π‘Ÿ for some π‘Ÿ > 1 . Then the following local weighted Poincare inequality for A-harmonic tensors was proved in [3]. There exists a constant 𝐢 independent of 𝑒 such that

ξ‚΅ 1 ξ€œ πœ‡ ( 𝐡 ) 𝐡 | | 𝑒 βˆ’ 𝑒 𝐡 | | 𝑠 ξ‚Ά 𝑑 πœ‡ 1 / 𝑠 | | 𝐡 | | ≀ 𝐢 1 / 𝑛 ξ‚΅ 1 | | 𝐡 | | ξ€œ 𝜎 𝐡 | | | | 𝑑 𝑒 𝑠 ξ‚Ά 𝑑 π‘₯ 1 / 𝑠 ( 2 . 2 ) for all balls 𝐡 with 𝜎 𝐡 βŠ‚ Ξ© , where the measure πœ‡ is defined by 𝑑 πœ‡ = 𝑀 ( π‘₯ ) 𝑑 π‘₯ , and 𝜎 𝐡 is the ball with the same center as 𝐡 and with d i a m ( 𝜎 𝐡 ) = 𝜎 d i a m ( 𝐡 ) .

3. Characterizations of Nonhomogeneous 𝐴 -Harmonic Equations

In what follows, we prove an A-harmonic analog of 1-harmonic equations.

Lemma 3.1. Let M be a complete noncompact Riemannian. For any π‘₯ 0 ∈ 𝑀 and any pair of positive numbers 𝑠 , 𝑑 with 𝑠 < 𝑑 , there exist a rotational symmetric Lipschitz continuous function πœ“ ( π‘₯ ) = πœ“ ( π‘₯ , 𝑠 , 𝑑 ) and a constant 𝑐 1 > 0 (independent of π‘₯ 0 , 𝑠 , 𝑑 ) with the following properties: (i) πœ“ ≑ 1 on B( π‘₯ 0 ; s) and πœ“ ≑ 0 off B ( π‘₯ 0 ; s), (ii) | βˆ‡ πœ“ | ≀ 𝑐 1 / ( 𝑑 βˆ’ 𝑠 ) , a.e. on M.

Proof. See Andreotti and Vesentini [6], Yau [7], and Karp [8].

Theorem 3.2. Let Ξ© be a domain in β„œ 𝑛 containing a ball 𝐡 ( π‘₯ 0 , π‘Ÿ ) of radius π‘Ÿ , centered at π‘₯ 0 , and let 𝑔 ∢ Ξ© β†’ β„œ be a continuous function with 𝑔 β‰₯ 0 , and 𝑐 = i n f π‘₯ ∈ 𝐡 ( π‘₯ 0 , π‘Ÿ / 2 ) 𝑔 ( π‘₯ ) . ( 3 . 1 ) Let 𝑓 ∢ Ξ© β†’ β„œ be a 𝐢 1 weak solution of ξ‚΅ d i v 𝐴 π‘₯ , βˆ‡ 𝑓 | | | | ξ‚Ά βˆ‡ 𝑓 = 𝑔 ( π‘₯ ) o n Ξ© , ( 3 . 2 ) then the infimum c satisfies 0 ≀ 𝑐 ≀ πœ† π‘˜ 2 𝑛 / π‘Ÿ , where k only depends on 𝑀 , πœ“ , and 𝑐 1 .

Proof. Let πœ“ β‰₯ 0 be as in Lemma 3.1, and 𝑀 = β„œ 𝑛 , 𝑑 = π‘Ÿ , 𝑠 = π‘Ÿ / 2 . Choose πœ“ to be a test function. Then by Cauchy-Schwarz inequality, we have ξ€œ 𝐡 ( π‘₯ 0 , π‘Ÿ / 2 ) ξ€œ 𝐢 πœ“ ( π‘₯ ) 𝑑 π‘₯ ≀ 𝐡 ( π‘₯ 0 , π‘Ÿ / 2 ) ξ€œ 𝑔 ( π‘₯ ) πœ“ ( π‘₯ ) 𝑑 π‘₯ ≀ 𝐡 ( π‘₯ 0 , π‘Ÿ ) βˆ’ ξ€œ 𝑔 ( π‘₯ ) πœ“ ( π‘₯ ) 𝑑 π‘₯ 𝐡 ξ€· π‘₯ 0 ξ€Έ , π‘Ÿ 𝐴 ξ‚΅ π‘₯ βˆ‡ 𝑓 | | | | ξ‚Ά ξ€œ βˆ‡ 𝑓 πœ“ ( π‘₯ ) 𝑑 π‘₯ ≀ 𝐡 ξ€· π‘₯ 0 ξ€Έ , π‘Ÿ | | | | πœ† 𝑀 ( π‘₯ ) βˆ‡ πœ“ 𝑑 π‘₯ ( 3 . 3 ) Hence, 𝑐 𝑉 π‘œ 𝑙 ( 𝐡 ( π‘₯ 0 , π‘Ÿ / 2 ) ) ≀ ( πœ† 𝑐 1 𝑐 2 ) / π‘Ÿ 𝑉 π‘œ 𝑙 ( 𝐡 ( π‘₯ 0 , π‘Ÿ ) ) .
Where 𝑐 2 only depends on 𝑀 and πœ“ .

Corollary 3.3. Let 𝑓 ∢ β„œ 𝑛 β†’ β„œ be a 𝐢 1 weak subsolution of 𝐴 -harmonic equation (1.1) with constant 1-tension field 𝑐 , that is, 0 ≀ d i v 𝐴 ( π‘₯ , βˆ‡ 𝑓 / | βˆ‡ 𝑓 | ) = 𝑐 in the distribution sense. Then 𝑓 is an A-harmonic function.

Remark. In a similar fashion, the above results can be extended to the nonhomogeneous equation d i v 𝐴 ( π‘₯ , βˆ‡ 𝑓 ) = 𝑔 ( π‘₯ ) by using Sobolev Imbedding Theorem.

4. Further Discussions

It would be interesting to find similar results of Section 2 for nonhomogeneous 𝐴 -harmonic equations. It would also be interesting to seek analogs of 1-harmonic applications in calibration geometry. The extension of 1-harmonic functions to 𝐴 -harmonic functions on hyperbolic spaces and their associated spaces could be explored.

To conclude this paper, we state another 𝐴 -harmonic extension of 1-harmonic result [5], that is an immediate consequence of Corollary 3.3.

Theorem. Let 𝑓 ∈ 𝐻 1 , 1 l o c ( 𝑅 𝑛 ) , and βˆ‡ 𝑓 β‰  0 for every x in 𝑅 𝑛 . The following statements are equivalent. (i) 𝑓 ∢ 𝑅 𝑛 β†’ 𝑅 is a 𝐢 1 weak subsolution of (1.1) with constant 1-tension field. (ii) 𝑓 is a 𝐢 1 weak solution of (1.1) on 𝑅 𝑛 . (iii) 𝑓 is a 𝐢 1 A-harmonic function on 𝑅 𝑛 .

Acknowledgments

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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