Liouville's theorem for degenerate elliptic and parabolic equations

1981 ◽  
Vol 29 (3) ◽  
pp. 203-209
Author(s):  
Yu. T. Kuz'menko
Keyword(s):  
Parabolic Equations ◽  
Elliptic And Parabolic Equations ◽  
Degenerate Elliptic ◽  
Liouville’S Theorem

scholarly journals Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

2018 ◽  
Vol 149 (2) ◽  
pp. 353-385
Author(s):  
Kin Ming Hui ◽  
Sunghoon Kim
Keyword(s):  
Growth Condition ◽  
Compact Subset ◽  
Parabolic Equations ◽  
Symmetric Solution ◽  
Blow Up ◽  
Radially Symmetric Solution ◽  
Singular Limits ◽  
Elliptic And Parabolic Equations ◽  
Degenerate Elliptic ◽  
Alpha Beta

AbstractLet n ⩾ 3, 0 ⩽ m < n − 2/n, ρ1 > 0, $\beta>\beta_{0}^{(m)}=(({m\rho_{1}})/({n-2-nm}))$, αm = ((2β + ρ1)/(1 − m)) and α = 2β+ρ1. For any λ > 0, we prove the uniqueness of radially symmetric solution υ(m) of Δ(υm/m) + αmυ + βx · ∇υ = 0, υ > 0, in ℝn∖{0} which satisfies $\lim\nolimits_{|x|\to 0|}|x|^{\alpha _m/\beta }v^{(m)}(x) = \lambda ^{-((\rho _1)/((1-m)\beta ))}$ and obtain higher order estimates of υ(m) near the blow-up point x = 0. We prove that as m → 0+, υ(m) converges uniformly in C2(K) for any compact subset K of ℝn∖{0} to the solution υ of Δlog υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn\{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$. We also prove that if the solution u(m) of ut = Δ (um/m), u > 0, in (ℝn∖{0}) × (0, T) which blows up near {0} × (0, T) at the rate $ \vert x \vert ^{-{\alpha_{m}}/{\beta}}$ satisfies some mild growth condition on (ℝn∖{0}) × (0, T), then as m → 0+, u(m) converges uniformly in C2 + θ, 1 + θ/2(K) for some constant θ ∈ (0, 1) and any compact subset K of (ℝn∖{0}) × (0, T) to the solution of ut = Δlog u, u > 0, in (ℝn∖{0}) × (0, T). As a consequence of the proof, we obtain existence of a unique radially symmetric solution υ(0) of Δ log υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn∖{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$.

Introduction

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Maximum Principle ◽  
Parabolic Equations ◽  
Theory Model ◽  
Degenerate Elliptic Operators ◽  
The Maximum Principle ◽  
Elliptic And Parabolic Equations ◽  
Degenerate Elliptic ◽  
Regularity Results ◽  
Manifolds With Corners

This book proves the existence, uniqueness and regularity results for a class of degenerate elliptic operators known as generalized Kimura diffusions, which act on functions defined on manifolds with corners. It presents a generalization of the Hopf boundary point maximum principle that demonstrates, in the general case, how regularity implies uniqueness. The book is divided in three parts. Part I deals with Wright–Fisher geometry and the maximum principle; Part II is devoted to an analysis of model problems, and includes degenerate Hölder spaces; and Part III discusses generalized Kimura diffusions. This introductory chapter provides an overview of generalized Kimura diffusions and their applications in probability theory, model problems, perturbation theory, main results, and alternate approaches to the study of similar degenerate elliptic and parabolic equations.

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