Abstract
Let be the unit disk in the complex plane and denote . Write for the class of all sense-preserving homeomorphism of onto the boundary of a convex Jordan domain . In this paper, five equivalent conditions for the solutions of triharmonic equations with Dirichlet boundary value conditions and to be Lipschitz continuous are presented.
1. Introduction
Let be the unit disk in the complex plane and denote . Let D and be subdomains of the complex plane Write for the set of all complex-valued n-times continuously differentiable functions from D into . A real-valued function u, defined in an open subset D of the complex plane , is harmonic if it satisfies Laplace’s equation:
A complex-valued function is harmonic if both u and are real harmonic. For a harmonic mapping f in it can be represented aswhere h and are analytic in and unique determined by . By a result of Lewy [1], the Jacobian of a sense-preserving locally univalent harmonic f in satisfies
See [2] for more properties of harmonic mappings in the plane. We say that a function is absolutely continuous on line in the region D if for every closed rectangle with sides parallel to the axes x and y, u is absolutely continuous on almost every horizontal line and almost every vertical line in R. Such a function has, of course, partial derivatives and everywhere in D. A sense-preserving topological mapping is said to be -quasiconformal if it satisfies:
f is absolutely continuous on lines in There are constants and , such thatwhere , , and If , then f is called a K-quasiconformal mapping. It is well known that a quasiconformal mapping f from onto a Jordan domain has a homeomorphic extension to the closure ; see [3].
A mapping h from D onto is said to be Lipschitz if there exists constant , such that the following inequalityholds for every .
Let and which satisfies the following triharmonic equation:with the following associated Dirichlet boundary value condition:where By [[4], Theorem 3.2.2], it is known that all solutions to equation (6) satisfying the condition (7) are given bywhere
1.1. Background
For a harmonic homeomorphism f of the unit disk onto itself, it is interesting to study the conditions which can imply f to be quasiconformal. It was Martio [5] who was the first one to pose and study for this problem. Let be the boundary function of and he proved that the conditions satisfy and the Dini conditionwhere ϖ is the modulus of continuity of ,can guarantee to imply the quasiconformality of . Pavlović [6] considered the above question to replace the Dini condition by the Hilbert transformation of belong to . In [7], Kalaj and Mateljević gave the equivalent conditions for a harmonic orientation preserving diffeomorphism between two plane Jordan domains and D with boundaries to be -quasiconformal. Their result was generalized to the case for -quasiconformal mappings satisfying Poisson’s equation by Li et al. [8]. In [9], Chen and Chen obtained several equivalent conditions for a harmonic mapping of upper half-plane onto itself to be -quasiconformal mapping. Recently, after characterizing the boundary conditions for Lipschitz continuity of -harmonic mappings, Chen [10] presented four equivalent conditions for the -quasiconformal solutions of the -Poisson equation with a nonhomogeneous term to be Lipschitz continuous. His result is a kind of generalization of the result obtained by Pavlović [6].
In what follows, we write for the class of all sense-preserving homeomorphism of onto the boundary of a convex Jordan domain . The aim of this paper is to study for the solutions of triharmonic equations with Dirichlet boundary value condition to be Lipschitz continuous.
The organization at the rest of this paper is as follows: In Section 2, we will make some estimations which will be used for proving the main result (Theorem 1). In Section 3, the proof of Theorem 1 is given. At the last section, an example is given to show that the family of solution (see this definition in Section 3) is nonempty.
2. Some Estimations
The following lemma was obtained in [11].
Lemma 1 [[11], Lemma 2.1]. Suppose that and are defined in (10). Then, the following inequalityholds for every where and
In order to give the complete proof of Theorem 1, we also need the following two lemmas.
Lemma 2. Suppose that and are defined in (11). Then, the following inequalitieshold for every , where .
Proof. Here we only calculate the estimation of because the calculation for another one is similar. LetwhereBy virtue of Hölder’s inequality, we getHence,This finishes the proof.
Lemma 3. Suppose that and are defined in (12). Then, the following inequalitieshold for every , where and
Proof. Here we only calculate the estimation of because the calculation for another one is similar. In order to do this, we introduce following functions:whereFirstly, we give the estimation of . SetThen,Therefore,whereSincewe get (by letting )which implies thatSecondly, we derive the estimation of . This can easily been obtained byThirdly, we calculate the upper bound of . By letting , we getThen,Next, we estimate the upper bound of . By letting , we getTherefore,At last, we calculate the upper bound of . By letting , we getHence,Combining with above estimations from to , we getwhere are defined in (15). This finished the proof.
3. Main Results
The Hilbert transform of a function is defined by the formulafor a.e. . See [[12], Chapter VII]. In what follows, we always say that is a convex Jordan domain with boundary. Let denote the family of solutions of the triharmonic equation in with the boundary value conditions and , , and each ω is a sense-preserving diffeomorphism. The following theorem gives a boundary characterizations of -quasiconformal mappings from onto satisfying triharmonic equations to be Lipschitz continuous.
Theorem 1. Suppose that with the representation (8). Then, the following conditions are equivalent:(a)ω is a -quasiconformal mapping and in , where is a constant(b)ω is Lipschitz with respect to Euclidean metric(c) is Lipschitz with respect to Euclidean metric(d) is a -quasiconformal mapping(e) is absolutely continuous on , and
Proof. Since , by a Theorem of Radó [13], Kneser [14], and Choquet [15], is sense-preserving harmonic diffeomorphism between and .
We now first prove that holds. By Lemmas 1–3, we haveHence, the proof of equivalence of (b) and (c) is derived by (43)-(45) and inequalities (46)Secondly, we prove that holds. By the -quasiconformality of ω, we getLet . Thus,Hence,By inequalities (43)–(45), we obtainTherefore, by (50)–(52), we getThis finishes the implication from to
Thirdly, we prove that holds. Since ω is Lipschitz and is equivalent to is Lipschitz. Thus, there are two constants M and L such that and hold for every Therefore,which says that is a -quasiconformal mapping. Moreover,Thus, the proof of implication from to is finished.
At last, since is a sense-preserving harmonic diffeomorphism, the equivalence for , and follows from [[7], Corollary 1.3]. Hence, the proof is finished.
Remark 1. The reason to suppose that in part of Theorem 1 can be seen [10, Proposition 1.1].
4. An Example
Following example says that the family of solution is nonempty.
Example 1. Let Then,
ω satisfies the triharmonic equation in and Dirichlet boundary value condition: ω is a -quasiconformal mapping of onto and in .
ω is Lipschitz continuous.
Proof. Let Obviously, , , , and . It follows thatThus, ω is a sense-preserving -quasiconformal homeomorphism from onto In addition, by (8) and (50)–(52), we getHence, we have proved that satisfies and
Further,which says that ω is Lipschitz continuous. Hence, the proof of is finished.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.