  Many physical problems can be modelled by partial differential equations on
unknown domains. Several examples can easily be found in the dynamics of free
interfaces in fluid dynamics, solid mechanics or in fluid-solid interactions.
To solve these equations in an arbitrary domain with nonlinear deformations, we
propose a mathematical approach allowing to track the boundary of the domain,
analogue of, and complementary to, the Arbitrary Lagrangian-Eulerian (ALE)
method for the interior of the domain. We name this method as the Boundary
Arbitrary Lagrangian-Eulerian (BALE) method. Additionally, in many situations
nonlinear deformations can be avoided with the help of some analyses which rely
on small deformations of the boundary, such as stability analysis, asymptotic
expansion and gradient-based shape optimisation. For these cases, we propose an
approach to perturb the domain and its boundaries and write the partial
differential equations at the unperturbed domain together with the boundary
conditions at the unperturbed boundary, instead of at the perturbed ones, which
are a priori unknown. We name this method as the Deformable Boundary
Perturbation (DBP) method. These two proposed methods rely on the boundary
exterior differential operator, whose relevant properties for the present work
are evidenced. We show an example for which the BALE and DBP methods are
applied, and for which we include the weak formulation revealing the
appropriateness of the finite element method in this context.
