# Well posedness for multi-dimensional junction problems with Kirchoff-type conditions

## TL;DR

This paper establishes the uniqueness of viscosity solutions for multi-dimensional junction problems with Kirchoff-type conditions, extending previous one-dimensional results and ensuring convergence of approximations without convexity assumptions.

## Contribution

It introduces the first comprehensive analysis of multi-dimensional junctions with Kirchoff conditions, proving uniqueness and convergence of viscosity solutions without convexity constraints.

## Key findings

- Uniqueness of viscosity solutions for multi-dimensional junctions.
- Convergence of viscosity-type approximations to a unique limit.
- Extension of one-dimensional junction results to higher dimensions.

## Abstract

We consider multi-dimensional junction problems for first- and second-order pde with Kirchoff-type Neumann boundary conditions and we show that their generalized viscosity solutions are unique. It follows that any viscosity-type approximation of the junction problem converges to a unique limit. The results here are the first of this kind and extend previous work by the authors for one-dimensional junctions. The proofs are based on a careful analysis of the behavior of the viscosity solutions near the junction, including a blow-up argument that reduces the general problem to a one-dimensional one. As in our previous note, no convexity assumptions and control theoretic interpretation of the solutions are needed.