We study existence questions and qualitative properties of solutions to variational problems related to minimization of geometric quantities, such as generalized notions of length of curves and of the area of surfaces, in a suitable sense. In the first part we consider a nonnegative multi-well potential function and we study the existence of heteroclinic orbits joining distinct minima of the potential, which are defined as classical solutions to a system of second order ordinary differential equations of gradient type. We find a geometric condition on the wells that guarantees the existence of heteroclinic connection between the wells. When this condition is met, the heteroclinic is built as a re-parametrization of a geodesic curve that minimizes a distance related to the potential, between the two wells. Next, we prove these heteroclinic orbits can be characterized as limits, in an appropriate sense, of sequences of periodic solutions to the system of ordinary differential equations previously considered, where each orbit joins the two connected components of the same level set of the potential, as the value of the level set approaches the global minimum.

In the second part we prove the existence and show regularity of functions that minimize an inhomogeneous version of the total variation functional on a fixed domain and subject to Dirichlet data, in arbitrary dimensions. Assuming, among other things, that the weight function is positive and the continuity of the Dirichlet data, we adapt a procedure developed by Sternberg-Williams-Ziemer to construct a continuous solution of this problem level set by level set, where each level set is a hypersurface minimizing an area related to the weight function, amongst competitors compatible with the boundary data.