Limiting observations for planar free-boundaries governed by isotropic-anisotropic singular diffusions, upper bounds for the limits

Abstract

In this paper, variational inclusions of Euler--Lagrange types, governed by two-dimensional isotropic-anisotropic singular diffusions, are considered. On that basis, we focus on the geometric structures of free boundaries where anisotropic conditions tend to isotropic. In this light, a limit-set of special piecewise-constant solutions will be presented. The objective in this paper is to give some observations on the upper bounds of the limit set with geometric characterizations. As a consequence, it will be shown that the isotropic free boundaries, as in the limit set, consist of a finite number of $ C^{1,1} $-Jordan curves, and these have certain geometric connections with the approaching anisotropic situations. Observations for the lower bounds will be studied in the sequel to this paper.