A compactness theorem for a new class of functions of bounded variation.

*(English)*Zbl 0767.49001With 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.

##### MSC:

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 |