scholarly journals The boundary behavior of the unique solution to a singular Dirichlet problem

2012 ◽  
Vol 391 (1) ◽  
pp. 278-290 ◽  
Author(s):  
Zhijun Zhang ◽  
Bo Li
Keyword(s):  
Dirichlet Problem ◽  
Unique Solution ◽  
Boundary Behavior ◽  
Singular Dirichlet Problem

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

scholarly journals Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

2021 ◽  
Vol 11 (1) ◽  
pp. 321-356
Author(s):  
Haitao Wan ◽  
Yongxiu Shi ◽  
Wei Liu
Keyword(s):  
Dirichlet Problem ◽  
Convex Domain ◽  
Boundary Behavior ◽  
Strictly Convex ◽  
Convex Solution ◽  
Regularly Varying ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

Abstract In 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 ( D 2 u ) = 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 ℝ N (N ≥ 2), b ∈ C∞(Ω) is positive and may be singular (including critical singular) or vanish on the boundary, g ∈ C 1((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $ and g is 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.

The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term

2006 ◽  
Vol 136 (1) ◽  
pp. 209-222 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Convection Term ◽  
Nonlinear Dirichlet Problem ◽  
Singular Dirichlet Problem

We show the existence and exact asymptotic behaviour of the unique solution u ∈ C2(Ω)∩C(Ω̄) near the boundary to the singular nonlinear Dirichlet problem −Δu = k(x)g(u) + λ|∇u|q, u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded domain with smooth boundary in RN, λ ∈ R, q ∈ [0, 2], g(s) is non-increasing and positive in (0, ∞), lims→0+g(s) = +∞, k ∈ Cα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.

On Dirichlet problem for Beltrami equations in finitely connected domains

2021 ◽  
Vol 34 ◽  
pp. 85-99
Author(s):  
Ihor Petkov ◽  
Vladimir Ryazanov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Boundary Behavior ◽  
Beltrami Equation ◽  
Elliptic Systems ◽  
Beltrami Equations ◽  
Prime Ends ◽  
Wide Circle ◽  
Continuous Boundary ◽  
Homeomorphic Solution

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.

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