Yang-Mills Theory on Fractal Manifolds with Boundary (Part B) Proof of the Existence of Minimizers for the Fractal Yang-Mills Energy Functional

This paper proves the existence of a minimal value for the Yang-Mills energyfunctional defined fractal manifolds under certain conditions. Fractal manifolds arecharacterized by non-integer Hausdorff dimensions, making their geometric proper?ties more complex than those of conventional integer-dimensional manifolds. Lever?aging Sobolev space theory and variational methods, we demonstrate that the en?ergy functional is bounded below in an appropriate function space. Utilizing thecompactness of minimizing sequences and weak lower semicontinuity, we constructa solution that satisfies the minimal value condition. The results of this study pro?vide a solid theoretical foundation for gauge field theory in the context of fractalgeometry, paving the way for investigations into quantum field theory and physicalphenomena (such as superconductivity and quantum gravity) on fractal manifolds.