Existence and uniqueness of a weak solution of an extensible beam on a moving domain
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Various methods exist to address the challenges posed by moving domain problems, including the transform method, penalty\linebreak 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 explore 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.
Abstract
Various methods exist to address the challenges posed by moving domain problems, including the transform method, penalty\linebreak 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 explore 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.