We develop the fundamental tools from functional analysis and partial differential equations to study the geometric and analytic aspects of triple junction hypersurfaces, a special class of singular manifolds whose boundaries are identified in a particular manner. We define some useful spaces on such singular objects and describe a kind of second-order elliptic operator defined on these function spaces. We extend the standard results in PDE theory for second-order elliptic operators on smooth Riemannian manifolds, including existence, regularity, spectrum theory, etc., to our singular setting. After that, we mention some applications of this theory, including the study of the Morse index for minimal hypersurfaces with triple junctions and the conformal structure on surfaces with triple junctions.
Our new PDE theory is essential to the study of immersed minimal hypersurfaces with triple junctions. In [Wang22], we have observed the appearance of such function spaces as an example. This motivates our study of these function spaces in a more general setting. In [Wan21a, Wan21b], we note that it is vital to have a regularity result so that we can use the powerful tools from elliptic PDEs. This is another motivation for the general theory of elliptic partial differential theory on triple junction hypersurfaces.
Once we have established the regularity, almost all PDE tools can be applied to triple junction hypersurfaces. In particular, we expect these results can also be extended to other geometric settings. For instance, we may consider defining heat-type equations on hypersurfaces with triple junctions. We may also consider more irregular hypersurfaces like surface clusters.