The variational cost of the change of direction in a planar curve and application to regularity of 2-dimensional sets

Abstract

We give an elementary proof for the well known regularity theorem of boundary with prescribed mean curvature [see Massari (Arch Rat Mech An 55:357–382, 1974), Massari (Rend Sem Mat Univ Padova 53:37–52, 1975)] for the case of planar subgraphs whose variational curvature belongs to $$L^p(\mathcal {R}^2), p>2$$ . The proof is based upon ideas already introduced in the precedent paper Barozzi (On sets with variational mean curvature in $$L^{n}(\mathcal{{R}}^{n})$$ , 2013). Finally, we give an example of a subgraph $$E$$ of a $$C^\infty $$ -function with $$H_E\in L^1(\mathcal {R}^2))$$ but $$H_E\notin L^p(\mathcal {R}^2), \forall p>1$$ .