Abstract
AbstractIn this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation det(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,$$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$where Ω is a bounded, smooth and strictly convex domain in ℝN(N≥ 2),b∈ C∞(Ω) is positive and may be singular (including critical singular) or vanish on the boundary,g ∈ C1((0, ∞), (0, ∞)) is decreasing on (0, ∞) withlimt→0+g(t)=∞$ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $andgis normalized regularly varying at zero with index −γ(γ>1). Our results reveal the refined influence of the highest and the lowest values of the (N − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.
Highlights
Introduction and main resultsThis presentation is to establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge–Ampère equation det(D u) = b(x)g(−u), u < in Ω and u = on ∂Ω, (1.1)where Ω is a bounded, smooth and strictly convex domain in RN(N ≥ ), and D u(x) =∂ u(x) ∂xi∂xj N×N denotes the Hessian of u and D u is the so called Monge–Ampère operator
Our results reveal the re ned in uence of the highest and the lowest values of the (N − )-th curvature on the second boundary behavior of the unique strictly convex solution to the problem
In Theorem 1.1, if Ω is a ball with radius R and center x, (I) When (S ) holds (or (S ) and (g )-(g ) hold with θ = in (g )), we have (i) If k ∈ Λ, the unique strictly convex solution u to problem (1.1) satis es
Summary
This presentation is to establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge–Ampère equation det(D u) = b(x)g(−u), u < in Ω and u = on ∂Ω, ∂ u(x) ∂xi∂xj N×N denotes the Hessian of u and D u is the so called Monge–Ampère operator. The nonlinearity g satis es (g ) g ∈ C (( , ∞), ( , ∞)) is decreasing on ( , ∞) and limt→ + g(t) = ∞; (g ) there exist a constant γ > and some function f ∈ C ( , a ) ∩ C[ , a ) for a su ciently small constant a > such that
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