AN EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION OF THE DIRICHLET PROBLEM WITH THE DATA IN MORREY SPACES

. Let 𝑛 − 2 < 𝜆 < 𝑛 , 𝑓 be a function in Morrey spaces 𝐿 1,𝜆 (Ω) , and the equation { be a Dirichlet problem, where Ω is a bounded open subset of ℝ 𝑛 , 𝑛 ≥ 3 , and 𝐿 is a divergent elliptic operator. In this paper, we prove the existence and uniqueness of this Dirichlet problem by directly using the Lax-Milgram Lemma and the weighted estimation in Morrey spaces.


Let be a bounded and open subset of
for 1 ≤ < ∞ and 0 ≤ ≤ . This Morrey spaces were introduced by C. B. Morrey [1] and still attracted the attention of many researcher to investigate its inclusion properties or application in partial differential equation [2,3,4,5,6,7,8].
Let ∈ 1, (Ω). In this paper, we will investigate the existence and uniqueness of the weak solution to the equation where is defined by (1) and the satisfies a certain condition. The Eq. (4) is called the Dirichlet problem.
Notice that, the result in [9], generalized by themselves in [12]. In [9,10,11], the authors used a representation of the weak solution, which involves the Green function [13], and proved that this representation satisfies (5) to show the existence of the weak solution of (4). Cirmi et. al [14] proved that the weak solution of (4) exists and unique, and its gradient belongs to some Morrey spaces, where they assumed ∈ 1, (Ω) for − 2 < < . The proof of the existence and uniqueness of the weak solution, which is done by Cirmi et. al, used an approximation method.
By assuming ∈ 1, (Ω), for − 2 < < , in this paper we will give a direct proof of the existence and uniqueness of the weak solution of the Dirichlet problem (4). Our method uses a functional analysis tool, i.e. the Lax-Millgram lemma, combining with a weighted embeddings in Morrey and Sobolev spaces.

RESEARCH METHODS
The constant = ( , , … , ), which appears throughout this paper, denotes that it is dependent on , , …, and . The value of this constant may vary from line to line whenever it appears in the theorems or proofs.
Our method relies on functional analysis tools, that is Lax-Milgram lemma, that we will state in this section. We start by write down some properties related to Lax-Milgram lemma.
Now we state the following two theorems regarding to the estimation for any functions in 0 1,2 (Ω), that we will need later. The first theorems called Poincar's inequality (see [15] for its proof) and the second theorem called sub representation formula (see [16] for its proof). We close this section by state the following Theorem which slightly modified from [17].
for every ∈ .

RESULTS AND DISCUSSION
To start our discussion, we prove that the bilinear mapping defined by (6) is continuous and coercive.

Lemma 2. The mapping defined by (6) is continuous and coercive.
Proof. Let ∈ 0 1,2 (Ω). We first prove the coercivity property. By using (3) and then Poincar's inequality, we have where = ( , ) is a positive constant. Now, we prove the continuity property. Let , ∈ 0 1,2 (Ω). Note that, according to (2). By using Hlder's inequality, we have This completes the proof.  We need the theorem below to prove that the function defined by (7) is a bounded linear functional. This theorem states about a weighted estimation in Morrey spaces where the weight in Sobolev spaces. The proof of this theorem was given in [11]. However, the given proof did not complete. Here we give the complete proof.
Notice that Hence The theorem is proved.  From Theorem 4, we obtain the following corollary. where the positive constant = ( , , , ‖ ‖ 1, (Ω) ). This means is also bounded and the proof is complete. 