2020/07/25 by Max Goering, Goering, Max
Mathematics · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2007.12953
A set of locally finite perimeter E ⊂ ℝn is called an anisotropic minimal surface in an open set A if Φ(E;A) ≤ Φ(F;A) for some surface energy Φ(E;A) = ∫∂*E ∩ A ‖ νE‖ d Hn-1 and all sets of locally finite perimeter F such that E ΔF ⊂ ⊂ A. In this short note we provide the details of a geometric proof verifying that all anisotropic surface minimizers in ℝ2 whose corresponding integrand ‖ ⋅ ‖ is strictly convex are locally disjoint unions of line segments. This demonstrates that, in the plane, strict convexity of ‖ ⋅ ‖ is both necessary and sufficient for regularity. The corresponding Bernstein theorem is also proven: global anisotropic minimizers E ⊂ ℝ2 are half-spaces.