A compactness theorem for a new class of functions of bounded variation. (English) Zbl 0767.49001

With the aim of solving some variational problems which arise in the theories of pattern recognition and of liquid crystals, the author introduces a class of generalized functions of bounded variation whose distributional derivatives are in a certain sense absolutely continuous with respect to the Lebesgue measure on \(\mathbb{R}^ n\) plus an \((n-1)\)- dimensional Hausdorff measure. The author then proves that functionals of a given type, which have many applications in the areas mentioned above, have minimisers in the class of generalised functions of bounded variation. The proof is based on a compactness theorem for this class of functions.


49J10 Existence theories for free problems in two or more independent variables
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
46A50 Compactness in topological linear spaces; angelic spaces, etc.
46F10 Operations with distributions and generalized functions