# Convexity and uniqueness in a free boundary problem arising in combustion theory.

• Volume: 17, Issue: 3, page 421-431
• ISSN: 0213-2230

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## Abstract

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We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain Ω ⊂ QT = Rn x (0,T) for some T &gt; 0 and a function u &gt; 0 in Ω such thatut = Δu,    in Ω,u = 0 and |∇u| = 1,   on Γ := ∂Ω ∩ QT,u(·,0) = u0,     on Ω0,where Ω0 is a given domain in Rn and u0 is a positive and continuous function in Ω0, vanishing on ∂Ω0. If Ω0 is convex and u0 is concave in Ω0, then we show that (u,Ω) is unique and the time sections Ωt are convex for every t ∈ (0,T), provided the free boundary Γ is locally the graph of a Lipschitz function and the fixed gradient condition is understood in the classical sense.

## How to cite

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Petrosyan, Arshak. "Convexity and uniqueness in a free boundary problem arising in combustion theory.." Revista Matemática Iberoamericana 17.3 (2001): 421-431. <http://eudml.org/doc/39654>.

@article{Petrosyan2001,
abstract = {We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain Ω ⊂ QT = Rn x (0,T) for some T &gt; 0 and a function u &gt; 0 in Ω such thatut = Δu,    in Ω,u = 0 and |∇u| = 1,   on Γ := ∂Ω ∩ QT,u(·,0) = u0,     on Ω0,where Ω0 is a given domain in Rn and u0 is a positive and continuous function in Ω0, vanishing on ∂Ω0. If Ω0 is convex and u0 is concave in Ω0, then we show that (u,Ω) is unique and the time sections Ωt are convex for every t ∈ (0,T), provided the free boundary Γ is locally the graph of a Lipschitz function and the fixed gradient condition is understood in the classical sense.},
author = {Petrosyan, Arshak},
journal = {Revista Matemática Iberoamericana},
keywords = {Problema de Dirichlet; Condiciones de contorno; Dominios convexos; Dominios de Lipschitz; Unicidad; Ecuación del calor; heat equation; propagation of flames; gradient condition},
language = {eng},
number = {3},
pages = {421-431},
title = {Convexity and uniqueness in a free boundary problem arising in combustion theory.},
url = {http://eudml.org/doc/39654},
volume = {17},
year = {2001},
}

TY - JOUR
AU - Petrosyan, Arshak
TI - Convexity and uniqueness in a free boundary problem arising in combustion theory.
JO - Revista Matemática Iberoamericana
PY - 2001
VL - 17
IS - 3
SP - 421
EP - 431
AB - We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain Ω ⊂ QT = Rn x (0,T) for some T &gt; 0 and a function u &gt; 0 in Ω such thatut = Δu,    in Ω,u = 0 and |∇u| = 1,   on Γ := ∂Ω ∩ QT,u(·,0) = u0,     on Ω0,where Ω0 is a given domain in Rn and u0 is a positive and continuous function in Ω0, vanishing on ∂Ω0. If Ω0 is convex and u0 is concave in Ω0, then we show that (u,Ω) is unique and the time sections Ωt are convex for every t ∈ (0,T), provided the free boundary Γ is locally the graph of a Lipschitz function and the fixed gradient condition is understood in the classical sense.
LA - eng
KW - Problema de Dirichlet; Condiciones de contorno; Dominios convexos; Dominios de Lipschitz; Unicidad; Ecuación del calor; heat equation; propagation of flames; gradient condition
UR - http://eudml.org/doc/39654
ER -

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