The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

Abstract

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

On the constancy theorem for anisotropic energies through differential inclusions

“…One therefore would like to show a priori that the multiplicity is constant and subsequently one is again in the situation given by (1.3)- (1.4). As for regularity theorems, no general constancy result is known at the moment for general functionals, except for the codimension one case, see [7].…”

On the constancy theorem for anisotropic energies through differential inclusions

“…. , X σi (5) ] is a T 5 configuration and moreover {C σ1(i) , C σ2(i) , C σ3(i) } are linearly independent for every i ∈ {1, . .…”

“…The definition of stationarity for geometric functionals is recalled in Section A. 5. In (4.11), Γ u is the graph of u, ξ u is its orientation, see (A.5), and θ(y) is a multiplicity, defined as…”

“…The matrices A, B, C, D appearing in Condition 3 are given by: These values fulfill Conditions 1, 2, 3. In particular, the three permutations in the definition of large T 5 configuration of Condition 2 are: [1,2,3,5,4], [1,2,4,5,3], [1,2,5,3,4].…”

“…One therefore would like to show a priori that the multiplicity is constant and subsequently one is again in the situation given by (1.3)- (1.4). As for regularity theorems, no general constancy result is known at the moment for general functionals, except for the codimension one case, see [5].…”