TY - JOUR
T1 - On the Monge-Ampère equation with boundary blow-up
T2 - Existence, uniqueness and asymptotics
AU - Cîrstea, Florica Corina
AU - Trombetti, Cristina
PY - 2008/2
Y1 - 2008/2
N2 - We consider the Monge-Ampère equation det D 2 u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on Ω. We assume that b C (overline)Ω is positive in Ω and non-negative on Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in R N with N 2. We give asymptotic estimates of the behaviour of such solutions near Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, (u) f (λ u)/f(u)=λ q, for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on Ω and b ≡ 0 on Ω.
AB - We consider the Monge-Ampère equation det D 2 u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on Ω. We assume that b C (overline)Ω is positive in Ω and non-negative on Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in R N with N 2. We give asymptotic estimates of the behaviour of such solutions near Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, (u) f (λ u)/f(u)=λ q, for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on Ω and b ≡ 0 on Ω.
UR - http://www.scopus.com/inward/record.url?scp=35548981366&partnerID=8YFLogxK
U2 - 10.1007/s00526-007-0108-7
DO - 10.1007/s00526-007-0108-7
M3 - Article
SN - 0944-2669
VL - 31
SP - 167
EP - 186
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 2
ER -