This paper deals with the following Dirichlet problem:

Let

Lipschitz continuity

uniform monotonicity

homogeneity

In particular, for

By Browder-Minty theory, see [

Given that

A-harmonic equations for differential forms have been a very active field in recent years because they are an invaluable tool to describe various systems of partial differential equations and to express different geometrical structures on manifolds. Moreover, they can be used in many fields, such as physics, nonlinear elasticity theory, and the theory of quasiconformal mappings, see [

This section is devoted to the notation of the exterior calculus and a few necessary preliminaries. For more details the reader can refer to [

We denote by

Let

Due to (

Suppose that

The notion of the generalized exterior coderivative

Suppose that

(i) Observe that generalized exterior derivatives have many properties similar to those of weak derivatives. For example,

(ii) If the generalized exterior derivative of

In fact, according to Definition

(iii) Together with the expression of differential forms, the definition of weak derivative and its uniqueness, we can prove that

(iv) Lastly, we refer to

A

The notion of vanishing normal part

For

Finally, we present briefly some spaces of differential forms:

We study the properties of solutions of the Dirichlet boundary value problem (

For each data

We start with a proposition which gives an important estimate for solutions of (

Given

Taking the solution

In this section, we establish the weak convergence of solutions of (

Under the hypotheses above,

To prove Theorem

One says that

It is easy to verify that it has the following equivalent definition.

One says that

According to the well-known results in Sobolev space in terms of functions, and together with the expression of differential forms and the diagonal rule we can easily obtain that

For

Suppose that a sequence of differential forms

On the one hand, since

On the other hand, since

Taking

In virtue of the fact that

For an arbitrary nonnegative test function

Note that (

Now, a Poincaré-Sobolev inequality for differential forms is needed.

Let

As a consequence of Lemma

Let

It follows from (

In conclusion, we summarize the above results in the following theorem.

For

This work was supported by the National Natural Science Foundation of China (Grant no. 11071048). The authors would like to express their sincere gratitude towards the reviewers for their efforts and valuable suggestions which greatly improved the paper.