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Limiting observations for planar free-boundaries governed by isotropic-anisotropic singular diffusions – upper bounds for the limits –. (English) Zbl 1259.35227

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
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Full Text: Euclid