A variational approach to a constrained hyperbolic Kirchhoff type problem with free boundary

In this study, we explore the existence of weak solutions for a hyperbolic Kirchhoff-type problem constrained by volume with a free boundary. Our approach leverages the hyperbolic discrete Morse flow. The presence of both a non-local term and a free boundary in our problem introduces significant challenges, necessitating unconventional methods for resolution. These unique difficulties are at the heart of our investigation. Specifically, the non-local term complicates the application of traditional analytical techniques, while the free boundary condition requires careful handling to ensure the existence of solutions. Our methodology involves a detailed examination of the hyperbolic discrete Morse flow framework, adapting it to address the intricacies introduced by the non-local term and the free boundary. Through rigorous mathematical analysis and the development of novel techniques, we aim to establish the conditions under which weak solutions exist.