On one-dimensional continua uniformly approximating planar sets

Miranda, Michele; Paolini, Emanuele; Stepanov, Eugene
November 2006
Calculus of Variations & Partial Differential Equations;Nov2006, Vol. 27 Issue 3, p287
Academic Journal
Consider the class of closed connected sets $$\Sigma\subset {\cal R}^n$$ satisfying length constraint $${\cal H}(\Sigma)\leq l$$ with given l>0. The paper is concerned with the properties of minimizers of the uniform distance F M of S to a given compact set $$M\subset {\cal R}^n$$ , where dist(y, S) stands for the distance between y and S. The paper deals with the planar case n=2. In this case it is proven that the minimizers (apart trivial cases) cannot contain closed loops. Further, some mild regularity properties as well as structure of minimizers is studied.


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