^{1}

^{1}

^{2}

^{1}

^{2}

In the paper, the necessary and sufficient condition of compact removability is obtained.

The questions of compact removability for Laplace equation is studied by Carleson [

Let

Denote by

Denote by

Let

The compact

The function

Let

Let

Analogously to

We will consider the elliptic operator in the bounded domain

The function

Here,

Let

We will assume that the coefficients of the operator

Let

Our aim to get the necessary and sufficient condition of compact removability

At first, we introduce some auxiliary statements.

If relative to the coefficients of the operator

Let relative to the coefficients of the operator

Let relative to the coefficients of the operator

Let relative to the coefficients of the operator

These lemmas are proved analogously to paper [

Let

There exists the unique function

It is easy to see that

It is clear that

The function

Let

It is sufficient to show the truth of the first part of assertion of lemma. Let

Lemma is proved.

Let

According to Lemma

The operator

By Lemma

By analogy with [

Let us say that the charge

At this, it is evident that

The weak solution

At first, we will show that if the function

We will obtain

Let

Then, from Lemma

Let now

The weak solution

In case

According to the above proved, if

For any charge of bounded variation on

Without losing generality, we will assume that the charge

Let us note that (

Let us consider now

According to Lemma

Let relative to coefficients of the operator

Without loss of generality, we can assume that the coefficients of the operator

Let

Let the conditions of the lemma and

Let relative to the coefficients of the operator

Let the ellipsoid

Analogously, the

According to the above proved, the function

Let relative to the coefficients of the operator

Let

Allowing for (

For showing the truth of the estimations from lower in (

Let

Allowing for (

Let relative to the coefficients of the operator

Upper estimation in (

If conditions (

Then,

If conditions (

If

For proving, at first, let us show that if

Let now for

Let

Further, let

The value

Let relative to the coefficients of the operator

We will use the following assertion, which is proved in [

Let the condition (

Let now

Let conditions of the real theorem be fulfilled and the compact

At first, let us note for proofing that the discussion are the same, as at the conclusion of estimation (

Further, analogously to Theorem

Hence, it follows that if mes

This paper was performed under the financial support of Science Foundation under the President of Azerbaijan.