Weakly differentiable functions on generalised submanifolds with mean curvature

In general dimensions, weak solutions of the Euler Lagrange equation of the area functional are naturally constructed in the class of integral varifolds. As the regularity theory of area-stationary integral varifolds is still substantially incomplete, it is important to enlarge the related mathematical machinery, both to handle the possible singularities and to advance regularity theory. In this regard, the talk will survey the implementation of the concept of weakly differentiable functions on integral varifolds of locally bounded first variation of area. The present theory of Sobolev spaces on metric measure spaces turns out to ill-adapted for this purpose. Instead, a coherent theory can be constructed by defining a non-linear class of functions taking into account the extrinsic geometry of the varifold. The inclusion of non-vanishing first variation therein is not only theoretically preferable but is also important for the needs of problems from applied analysis.