We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when a convex obstacle is present.

The distance function in the presence of an obstacle / Albano P.; Basco V.; Cannarsa P.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 61:1(2022), pp. 13.1-13.26. [10.1007/s00526-021-02125-z]

The distance function in the presence of an obstacle

Albano P.;
2022

Abstract

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when a convex obstacle is present.
2022
The distance function in the presence of an obstacle / Albano P.; Basco V.; Cannarsa P.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 61:1(2022), pp. 13.1-13.26. [10.1007/s00526-021-02125-z]
Albano P.; Basco V.; Cannarsa P.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/901222
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