A convex relaxation method for free boundary problems
Guy Bouchitté, Minh Phan · Evolution equations and control theory · 2025
We derive a convex relaxation principle for a large class of non convex variational problems where the functional to be minimized involves a one homogeneous gradient energy. This applies directly to free boundary or multiphase problems in the case of the classical total variation or of some anisotropic variants. The underlying argument is an exclusion principle which states that any global minimizer avoids taking values in the intervals where the lower order potential is nonconvex. This allows using duality methods and deriving a saddle point characterization of the global minimizers. A numerical validation of our principle is presented in the case of several free boundary and multiphase problems that we treat through a primal-dual algorithm. The accuracy of the interfaces and the convergence of the algoritm benefit in a large way of a new epigraphical projection method that we introduced to tackle the non differentiability of the convexified Lagrangian.