ABP inequalities for singular submanifolds of bounded mean curvature
Mario Santilli
TL;DR
This paper extends ABP inequalities to singular submanifolds with bounded mean curvature, providing new estimates and generalizing previous results for minimal surfaces.
Contribution
It introduces a curvature notion for arbitrary closed sets and proves ABP-type and weak-Harnack estimates for singular submanifolds.
Findings
Established ABP inequalities for singular submanifolds of arbitrary codimension
Derived weak-Harnack estimates from the ABP inequalities
Generalized results from minimal surface equations to broader classes of submanifolds
Abstract
Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous results by O. Savin for viscosity solutions of the minimal surface equation.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Numerical methods in inverse problems