General Theory of Partial Differential Equations on Triple Junction Surfaces

Sat, 02 Jul 2022 11:51:14 +0800

Main contributions

The thesis builds a general analytic framework for triple-junction hypersurfaces: singular spaces obtained by identifying the boundaries of several smooth sheets.

Function spaces

Analysis that remembers the junction geometry

Sobolev- and Hölder-type spaces are defined sheetwise with compatibility and balancing conditions along the common boundary, providing the correct domains for geometric variation problems.

Elliptic theory

Classical PDE tools extend to the singular setting

For second-order elliptic operators on triple-junction hypersurfaces, the thesis develops existence, regularity, Fredholm, and spectral results analogous to those on smooth Riemannian manifolds.

Applications

Geometry becomes accessible through the PDE framework

The theory is applied to Jacobi operators, Morse index questions for minimal triple junctions, and conformal and uniformization problems for singular surfaces.

Analytic viewpoint