Viscosity Solutions on Ramified Spaces

The thesis develops a viscosity solution theory for Hamilton–Jacobi equations on ramified spaces, particularly networks and graphs, and connects this with vanishing viscosity and the distance function.

Motivation & Goal

In many applications (e.g. interactions between different “branches”/media), PDEs are not defined on smooth manifolds but on spaces composed of several “branches” joined at transition points. For fully nonlinear equations (especially Hamilton–Jacobi), a rigorous viscosity solution theory for such settings was lacking.

Chapters 2–3: Foundations & Eikonal

Historical and conceptual background (Kruzhkov, Crandall–Lions) and a detailed analysis of the vanishing viscosity method applied to the eikonal problem. It is shown that on classical domains this method converges to the distance function to the boundary.

Chapter 4: Vanishing Viscosity on Networks

Extension to networks: for the second-order viscous regularization, one additionally requires a Kirchhoff condition (flux/derivative balance) at transition nodes in order to obtain uniqueness. Building on this, convergence results for the vanishing viscosity method on networks are derived.

Chapter 5: Viscosity Solutions on Networks (Main Contribution)

An intrinsic test-function-based definition of viscosity sub- and supersolutions on networks is formulated, including correct transition conditions at branching nodes. The key insight is: the limit obtained via vanishing viscosity does not satisfy the Kirchhoff condition, but rather a “pairwise” transition logic (evaluated over two incident edges at a time) – and this yields the comparison principle, uniqueness, and existence.

Chapter 6: Distance Function & Topology

Investigation of the distance function on networks: a “curvature functional” (counting suitably weighted local maxima/singularities of the distance function) is linked to a purely graph-theoretic/topological quantity of the network (including in the context of its cycle structure).

Chapter 7: Higher-Dimensional Ramified Spaces / LEP-Spaces

Introduction of ramified manifolds and the class of LEP-spaces (locally elementary polygonal ramified spaces), and extension of the network theory to these spaces. A complete existence theory for viscous regularizations is difficult in this geometry; however, convergence is shown provided a viscous approximation family exists.

Summary

The thesis provides a consistent framework for treating Hamilton–Jacobi equations on ramified spaces via viscosity solutions – including correct transition conditions, existence and uniqueness via the comparison principle, and consistency with vanishing viscosity. In addition, the distance function is characterized topologically as a structure-determining object.