The recent development in theories with more than three spatial dimensions has shifted from very small compact extra dimensions (Kaluza-Klein picture) towards a brane world picture, where the observable 3+1 dimensional world is located inside a domain wall - see the review paper by V. Rubakov, hep-ph/0104152.

Among many points of attraction of this approach, we mention the possibility to lower considerably the Planck scale of quantum gravity effects, to find solutions of fermion mass hierarchy in the Standard Model and to predict new physical phenomena for the next generation of high-energy colliders indicating the presence of extra dimensions.

In spite of the exploding number of publications in this subject, some basic and difficult problems remain nowadays unsolved.

As particular examples, we mention:

1) the natural implementation on the domain wall of light but massive fermions together with light massive scalar fields of Higgs type and light gauge fields;

2) the self-consistent mechanisms to create a thin domain wall;

3) the way to provide several domain walls on a controllable distance as it is needed in the scenarios supporting the fermion mass hierarchies.

The above problems must be solved not only in the flat space-time with extra dimensions but also in the Anti-de Sitter space and its relatives. We propose to investigate the above problems and to develop a self-consistent description of domain wall formation based on local 4-fermion interaction and other types of interaction of proto-particles living in the extra dimensions. The preliminary calculation has shown that this approach is visible, viable and predictive for the mass spectrum and the Yukawa coupling constants of fermions and bosons trapped by a domain wall. We also plan to analyze the similar mechanisms in 3+1 and lower dimensions in a search for new physical phenomena in the condensed matter physics.