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We consider a family of variational problems involving generalized Gauss graphs. Roughly, we are interested in the problem of minimizing a functional defined in a class of surfaces constrained to include a given fixed rectifiable set. As a particular case, one has the following example: Given a rectifiable set M , find a mass minimizer among all null boundary generalized Gauss graphs of surfaces which include M as a subset. Particular attention is paid to the case of curves in the plane.

Minimizing functionals depending on surfaces and their curvatures: A class of variational problems in the setting of generalized Gauss graphs