Advances in Differential Equations

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

Ken Shirakawa

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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.

Article information

Source
Adv. Differential Equations Volume 18, Number 3/4 (2013), 351-383.

Dates
First available in Project Euclid: 5 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1360073020

Mathematical Reviews number (MathSciNet)
MR3060199

Zentralblatt MATH identifier
1259.35227

Subjects
Primary: 35J75: Singular elliptic equations 35R35: Free boundary problems 14H50: Plane and space curves

Citation

Shirakawa, Ken. Limiting observations for planar free-boundaries governed by isotropic-anisotropic singular diffusions, upper bounds for the limits. Adv. Differential Equations 18 (2013), no. 3/4, 351--383. https://projecteuclid.org/euclid.ade/1360073020.


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