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\begin{document}

\title{An alternative potential method for mixed steady state elastic
oscillation problems}



\author[1]{David Natroshvili}%
\author[2]{Maia Mrevlishvili}%
\author[1]{Zurab Tediashvili}%
\affil[1]{Georgian Technical University}%
\affil[2]{Ilia State University}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\selectlanguage{english}
\begin{abstract}
We consider an alternative approach to investigate three-dimensional
exterior mixed boundary value problems (BVP) for the steady state
oscillation equations of the elasticity theory for isotropic bodies. The
unbounded domain occupied by an elastic body,  - [?] R 3 , has a compact
boundary surface S = [?]  - , which is divided into two disjoint parts,
the Dirichlet part S D and the Neumann part S N , where the displacement
vector (the Dirichlet type condition) and the stress vector (the Neumann
type condition) are prescribed respectively. Our new approach is based
on the classical potential method and has several essential advantages
compared with the existing approaches. We look for a solution to the
mixed boundary value problem in the form of a linear combination of the
single layer and double layer potentials with densities supported on the
Dirichlet and Neumann parts of the boundary respectively. This approach
reduces the mixed BVP under consideration to a system of boundary
integral equations, which contain neither extensions of the Dirichlet or
Neumann data nor the Steklov-Poincar\selectlanguage{ngerman}é type operator involving the
inverse of a special boundary integral operator, which is not available
explicitly for arbitrary boundary surface. Moreover, the right-hand
sides of the resulting boundary integral equations system are
vector-functions coinciding with the given Dirichlet and Neumann data of
the problem in question. We show that the corresponding matrix integral
operator is bounded and coercive in the appropriate L 2 -based Bessel
potential spaces. Consequently, the operator is invertible, which
implies unconditional unique solvability of the mixed BVP in the class
of vector-functions belonging to the Sobolev space {[} W 2 , loc 1 ( \selectlanguage{english} \selectlanguage{english}-
) {]} 3 and satisfying the Sommerfeld-Kupradze radiation conditions at
infinity. We also show that the obtained matrix boundary integral
operator is invertible in the L p -based Besov spaces and prove that
under appropriate boundary data a solution to the mixed BVP possesses C
\selectlanguage{greek}α -\selectlanguage{english}H\selectlanguage{ngerman}ölder continuity property in the closed domain \selectlanguage{english} \selectlanguage{english}- \selectlanguage{english}- with \selectlanguage{greek}α \selectlanguage{english}= 1 2 -
\selectlanguage{greek}ε , \selectlanguage{english}where \selectlanguage{greek}\emph{ε\textgreater{}}\selectlanguage{english}0 is an arbitrarily small number.%
\end{abstract}%



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