Shirakawa, Ken Limiting observations for planar free-boundaries governed by isotropic-anisotropic singular diffusions – upper bounds for the limits –. (English) Zbl 1259.35227 Adv. Differ. Equ. 18, No. 3-4, 351-383 (2013). Summary: 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. MSC: 35R35 Free boundary problems for PDEs 35J75 Singular elliptic equations 14H50 Plane and space curves PDFBibTeX XMLCite \textit{K. Shirakawa}, Adv. Differ. Equ. 18, No. 3--4, 351--383 (2013; Zbl 1259.35227) Full Text: Euclid