On Nonhomogeneous -Harmonic Equations and 1-Harmonic Equations

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 -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 function is said to be A-harmonic if it is a weak solution of A-harmonic equation

(1.1)

where is the length of the gradient of , and for a function without a critical point, is said to be the 1-tension field of .

Let be an open subset of . Consider the following second-order divergence-type elliptic equation:

(1.2)

where x satisfies

(i),

(ii),

where are two fixed constants, and is called a weight if and a.e. Also, in general where 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 -Harmonic Equations

Definition.

A function , is called a very weak solution of (1.2) if

(2.1)

for all 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.

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 [3]. There exists a constant independent of such that

(2.2)

for all balls with , where the measure is defined by , and is the ball with the same center as and with .

3. Characterizations of Nonhomogeneous -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 and any pair of positive numbers with , there exist a rotational symmetric Lipschitz continuous function and a constant (independent of ) with the following properties:

(i) on B(; s) and off B (; s),

(ii), a.e. on M.

Proof.

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

Theorem 3.2.

Let be a domain in containing a ball of radius , centered at , and let be a continuous function with , and

(3.1)

Let be a weak solution of

(3.2)

then the infimum c satisfies , where k only depends on , and .

Proof.

Let be as in Lemma 3.1, and . Choose to be a test function. Then by Cauchy-Schwarz inequality, we have

(3.3)

Hence, .

Where only depends on and .

Corollary 3.3.

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.

Remark.

In a similar fashion, the above results can be extended to the nonhomogeneous equation by using Sobolev Imbedding Theorem.

4. Further Discussions

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 [5], that is an immediate consequence of Corollary 3.3.

Theorem.

Let , and for every x in . The following statements are equivalent.

(i) is a weak subsolution of (1.1) with constant 1-tension field.

(ii) is a weak solution of (1.1) on .

(iii) is a A-harmonic function on .