Abstract
In the paper we consider three-dimensional Riquier-type and classical mixed boundary value problems for the polymetaharmonic equation
A Properties of potentials
For the readers’ convenience, we collect here some results describing the properties of the layer potentials. For the potentials associated with the Helmholtz equation the following theorems hold; see [1, 6, 12, 11, 26, 21, 23].
Let
Let
Let
Proof.
First we note that the operator
Let us show that the operator
Write Green’s formulae in the domains
where
Since
whence
Therefore the pseudodifferential operator
Analogously, we can prove the invertibility of the operator
The operator
is invertible for all
For the potentials of the polymetaharmonic equation (2.8) the following theorems hold.
Let
Let
Proof.
Substituting the fundamental solution (2.14) of the polymetaharmonic equation (2.8) in Green’s formula (5.1) instead of v, we can derive a representation formula for a function
and
where
From the asymptotic properties of the fundamental solution H near the origin it follows that,
for
Let
where
Denote
Then we can rewrite formulae (A.5) and (A.6) as
Therefore using equalities (3.2), (3.3) and (A.7), we derive the following relations:
Hence relation (A.3) holds.
Let us introduce the notations
where
Note that
are bounded for all
are mutually conjugate, i.e.
Let
Proof.
Let
Denote
Then we can rewrite formulae (A.5) and (A.6) as
Therefore using equalities (3.1), (3.3) and (A.7), we derive the relations
Hence relation (A.8) follows.
Further, using equalities (3.1), (3.3), (A.4) and (A.7), we derive the relations
Therefore (A.9) holds. ∎
The following assertions hold.
The operators
are continuous for all
The operators
are continuous for all
B Auxiliary propositions
The fundamental solution of the differential operator
The difference
Let us introduce the layer potentials
where
In view of estimates (B.1) and (B.2) it follows that the principal parts of the operators
are compact for all
is compact, where
The operator
is invertible.
Proof.
Since
Let
Introduce the function
Then (cf. Theorems 3.2, A.6 and A.7)
Therefore v is a solution of the homogeneous Dirichlet problem
It follows from the uniqueness of the solution of the Dirichlet problem (B.3)–(B.4) that
we arrive at the equations
Taking into account (B.5) and (B.6) in Green’s formula (2.5), we obtain the equality
Hence
is invertible. ∎
The operator
is a positive-definite operator, i.e.
where the symbol
Proof.
Let
where
Therefore for some positive constants
The operator
is a positive-definite operator, i.e.
where the symbol
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