# Non-Trapping sets and Huygens Principle

• Volume: 33, Issue: 3, page 517-530
• ISSN: 0764-583X

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## Abstract

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We consider the evolution of a set $\Lambda \subset {ℝ}^{2}$ according to the Huygens principle: i.e. the domain at time t>0, Λt, is the set of the points whose distance from Λ is lower than t. We give some general results for this evolution, with particular care given to the behavior of the perimeter of the evoluted set as a function of time. We define a class of sets (non-trapping sets) for which the perimeter is a continuous function of t, and we give an algorithm to approximate the evolution. Finally we restrict our attention to the class of sets for which the turning angle of the boundary is greater than -π (see [2]). For this class of sets we prove that the perimeter is a Lipschitz-continuous function of t. This evolution problem is relevant for the applications because it is used as a model for solid fuel combustion.

## How to cite

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Benedetto, Dario, Caglioti, Emanuele, and Libero, Roberto. "Non-Trapping sets and Huygens Principle." ESAIM: Mathematical Modelling and Numerical Analysis 33.3 (2010): 517-530. <http://eudml.org/doc/197469>.

@article{Benedetto2010,
abstract = { We consider the evolution of a set $\Lambda\subset \mathbb R^2$ according to the Huygens principle: i.e. the domain at time t>0, Λt, is the set of the points whose distance from Λ is lower than t. We give some general results for this evolution, with particular care given to the behavior of the perimeter of the evoluted set as a function of time. We define a class of sets (non-trapping sets) for which the perimeter is a continuous function of t, and we give an algorithm to approximate the evolution. Finally we restrict our attention to the class of sets for which the turning angle of the boundary is greater than -π (see [2]). For this class of sets we prove that the perimeter is a Lipschitz-continuous function of t. This evolution problem is relevant for the applications because it is used as a model for solid fuel combustion. },
author = {Benedetto, Dario, Caglioti, Emanuele, Libero, Roberto},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Eikonal equation; distance function.; eikonal equation; distance function; solid fuel combustion},
language = {eng},
month = {3},
number = {3},
pages = {517-530},
publisher = {EDP Sciences},
title = {Non-Trapping sets and Huygens Principle},
url = {http://eudml.org/doc/197469},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Benedetto, Dario
AU - Caglioti, Emanuele
AU - Libero, Roberto
TI - Non-Trapping sets and Huygens Principle
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 3
SP - 517
EP - 530
AB - We consider the evolution of a set $\Lambda\subset \mathbb R^2$ according to the Huygens principle: i.e. the domain at time t>0, Λt, is the set of the points whose distance from Λ is lower than t. We give some general results for this evolution, with particular care given to the behavior of the perimeter of the evoluted set as a function of time. We define a class of sets (non-trapping sets) for which the perimeter is a continuous function of t, and we give an algorithm to approximate the evolution. Finally we restrict our attention to the class of sets for which the turning angle of the boundary is greater than -π (see [2]). For this class of sets we prove that the perimeter is a Lipschitz-continuous function of t. This evolution problem is relevant for the applications because it is used as a model for solid fuel combustion.
LA - eng
KW - Eikonal equation; distance function.; eikonal equation; distance function; solid fuel combustion
UR - http://eudml.org/doc/197469
ER -

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