Research Article | Open Access
Some Properties of Solutions to a Class of Dirichlet Boundary Value Problems
This paper deals with the following Dirichlet problem: in on . Based on its solvability, we derive some properties of its solutions. In this paper, we mainly get three results. Firstly, we establish an integral estimate for the solutions of the above Dirichlet boundary value problem. Secondly, a stability result of solutions for varying differential forms and is obtained. Lastly, we present a weak reverse Hölder inequality for solutions.
Let be a bounded domain in with smooth boundary . Throughout this paper we assume that , , is a Hölder conjugate pair, . First, we consider the nonhomogeneous A-harmonic equation: where and is a Carathéodory mapping satisfying the following assumptions for fixed :(1)Lipschitz continuity (2)uniform monotonicity (3)homogeneity for almost every and all , .
In particular, for , then (1.1) is simplified to the nonhomogeneous -harmonic equation:
By Browder-Minty theory, see , the existence and uniqueness of a solution to the Dirichlet problem in has been obtained by Iwaniec et al., see . For the solution with vanishing tangential component on , we write it as .
Definition 1.1. Given that , a differential form is called a solution to the Dirichlet problem (1.6) if and it holds for all .
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 [3–10]. The purpose of this paper is to study properties of solutions of the Dirichlet boundary value problem (1.6) based on the existence of its solutions.
2. Notation and Preliminary Results
We denote by the space of -covectors in and the direct sum is a graded algebra with respect to the wedge product . We will make use of the exterior derivative: and its formal adjoint operator known as the Hodge codifferential, where the symbol denotes the Hodge star duality operator. Note that each of the operators and applied twice gives zero.
Let be the class of infinitely differentiable -forms on . Since is a smooth domain, near each boundary point one can introduce a local coordinate system such that on and such that the -curve is orthogonal to . Near this boundary point, every differential form can be decomposed as , where are called the tangential and the normal parts of , respectively. Now, the duality between and is expressed by the integration by parts formula for all and , provided or . The symbol denotes the inner product, that is, let and , then .
Due to (2.5), extended definitions for and can be introduced as the introduction of weak derivatives.
Definition 2.1 (see ). Suppose that and . If for every test form , one says that has generalized exterior derivative and write .
The notion of the generalized exterior coderivative can be defined analogously.
Definition 2.2 (see ). Suppose that and . If for every test form , one says that has generalized exterior coderivative and write .
Remark 2.3. (i) Observe that generalized exterior derivatives have many properties similar to those of weak derivatives. For example, if it exists, it is unique; if is differentiable in the conventional sense, then its generalized exterior derivative is identical to its the classical exterior differential . Analogous results hold for generalized exterior coderivative.
(ii) If the generalized exterior derivative of , , exists, then also has its generalized exterior derivative . Moreover, .
In fact, according to Definition 2.1, it holds for every test form . Thus, for every , we have and by taking in the above integral equality implies Therefore, Definition 2.1 yields that exists and . Similarly, we have .
(iii) Together with the expression of differential forms, the definition of weak derivative and its uniqueness, we can prove that and have analogous expressions, that is, for , we have where denotes the ordinary derivative and the weak derivative. So in the next we use to represent the action instead of , similar for and .
(iv) Lastly, we refer to as the closed -forms and to as the coclosed -forms.
Definition 2.4 (see ). A -form is said to have vanishing tangential component at in a generalized sense, if both and belong to and holds for any . One writes .
The notion of vanishing normal part can be defined analogously. Now, the following extension of the identity (2.5) can be introduced, by an approximation argument proved by Iwaniec and Lutoborski .
Proposition 2.5 (see ). For a Hölder conjugate pair and , one has provided or .
Finally, we present briefly some spaces of differential forms: —the space of -forms with coefficients in ; —the space of -forms such that is a regular distribution of ; —the Sobolev space of -forms defined by ; —the space of -forms in with vanishing tangential component on . which are used throughout this paper.
3. Main Results
We study the properties of solutions of the Dirichlet boundary value problem (1.6) whose existence can be deduced by the following.
3.1. An Integral Estimate
We start with a proposition which gives an important estimate for solutions of (1.6).
Proposition 3.2. Given , suppose that is a solution of (1.6) in . Then one has
Proof. Taking the solution as the test function in (1.7) yields thus we have Next we apply Young’s inequalityas follows: It follows from the structural assumptions (1) and (2) that By choosing and using again Young’s inequality, we have Therefore, we get that Finally, we obtain by choosing that The theorem follows.
3.2. Stability of Solutions
In this section, we establish the weak convergence of solutions of (1.6) with varying differential forms and . Particularly, given a sequence , Lemma 3.1 implies that there exists solution to Suppose that in , and in , we will show that is a solution of (1.6).
Theorem 3.3. Under the hypotheses above, is a solution of (1.6).
To prove Theorem 3.3, we need firstly to give the definition of the weak convergence for sequences in spaces of differential forms.
Definition 3.4. One says that is weakly convergent to in if
whenever and write in .
It is easy to verify that it has the following equivalent definition.
Definition 3.5. One says that weakly convergent to in if whenever and write in .
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
Proposition 3.6. For , is reflexive.
Proposition 3.7. in if and only if in and in , where and the partial differentiation is applied to the coefficients of .
Lemma 3.8. Suppose that a sequence of differential forms converges to weakly in while the generalized exterior derivatives of , , exist and stay bounded in . Then the generalized exterior derivative of , , exists and in .
Proof. On the one hand, since has generalized exterior derivative , then according to Definition 2.1 we have
for every . Notice that in and , it follows from Definition 3.4 that
Thus, we obtain
for every .
On the other hand, since stays bounded in , then it follows from Proposition 3.6 that there exists weakly convergence subsequence of , we may assume that in . Hence, for we have Combining (3.14) and (3.16), the uniqueness of the limit yields for every . Thus, Definition 2.1 implies that the generalized exterior derivative of exists and . Furthermore, we have from (3.15) that in . The lemma follows.
Proof of Theorem 3.3. Taking as test differential form in (1.7) for both then we obtain or, equivalently, we have Write and , then the above identity can be simplified to Then it follows from the Lipschitz condition (1), the monotonicity condition (2), and Hölder inequality that: Therefore, we obtain Since and are Cauchy sequences in and , respectively, we have and for all , and further we get by applying Proposition 3.2 that Combining (3.23)-(3.24), we obtain as . Thus, the Minkowski inequality implies as , that is, is a Cauchy sequence in . Thus, we have is bounded in . Then, it follows from Lemma 3.8 that exists and in . Note that is a Cauchy sequence in , then we have by the uniqueness of weak limit that in . According to Definition 2.4, we know that for any . And it follows easily that since in and is bounded. Hence, we have . The final step in our proof is to show that satisfies (1.7). Recall that for all . It is rather easy to obtain the following: as , that is, it holds as . On the other hand, an easy computation gives that as . Therefore, we obtain for all , which implies the desired result.
3.3. Weak Reverse Hölder Inequality
For an arbitrary nonnegative test function , we multiply (3.35) for and by the homogeneity assumption (3), and we obtain that Since we have Applying the operator to the above equation leads to a new nonhomogeneous A-harmonic equation as follows: where Then by applying Proposition 3.2 to (3.41) we have which implies Therefore, we obtain For an arbitrary cube , we can choose a cut-off function such that , and on , where denotes the Lebesgue measure of . Now inequality (3.45) yields
Note that (3.35) is not affected if a closed form is subtracted from and a coclosed form is subtracted from . Therefore, the above calculation shows
Now, a Poincaré-Sobolev inequality for differential forms is needed.
Lemma 3.9 (see ). Let be a cube in . Suppose that and , where , . Then there exist a closed form and a coclosed form such that
Corollary 3.10. Let be a cube in . Suppose that and , where , . Then there exist a closed form and a coclosed form such that where denotes the integral mean over , that is,
In conclusion, we summarize the above results in the following theorem.
Theorem 3.11. For a Hölder conjugate pair. Given , suppose that is a solution of (1.6). Then one has where denotes the integral mean over , that is,
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.
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Copyright © 2012 Tingting Wang and Gejun Bao. 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.