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.
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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 -