Journal of the
Korean Mathematical Society JKMS

Various methods exist to address the challenges posed by moving domain problems, including the transform method, penalty method, and push-forward and pull-back method.
These methods play a crucial role in transforming non-cylindrical domain problems into cylindrical domain problems.
In \cite{RINCON2007}, the existence of at least one weak solution for a more general extensible beam was established.
However, this paper does not specifically address the uniqueness of solutions and exploring their relationship within any finite-dimensional space.
Our primary objective is to formulate approximate solutions in a finite-dimensional space using the penalty method,
with a focus on achieving uniqueness. Furthermore, we present an illustrative example to enhance the understanding
of the concepts discussed.

Keywords: Shallow arch, extensible beam, moving domain, weak solution

MSC numbers: 47J35, 35Q74, 35C05, 70G75