TY - JOUR

T1 - The minimum speed for a blocking problem on the half plane

AU - Bressan, Alberto

AU - Wang, Tao

N1 - Funding Information:
This work was supported by NSF through grant DMS-0807420, “New problems in nonlinear control”. The authors also wish to thank Camillo De Lellis for useful suggestions.

PY - 2009/8/1

Y1 - 2009/8/1

N2 - We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R2 be a compact, simply connected set with smooth boundary. We define dK (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then dK attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

AB - We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ R2 be a compact, simply connected set with smooth boundary. We define dK (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then dK attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.

UR - http://www.scopus.com/inward/record.url?scp=63349108339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63349108339&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.02.039

DO - 10.1016/j.jmaa.2009.02.039

M3 - Article

AN - SCOPUS:63349108339

SN - 0022-247X

VL - 356

SP - 133

EP - 144

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -