scholarly journals Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
M. E. Amendola ◽  
L. Rossi ◽  
A. Vitolo
Keyword(s):  
Maximum Principle ◽  
Type Theorem ◽  
Unbounded Domains ◽  
Local Maximum ◽  
Maximum Principles ◽  
Gradient Term ◽  
Harnack Inequalities ◽  
General Unbounded Domains ◽  
Second Order Elliptic Equations ◽  
Weak Harnack Inequality

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

Discrete maximum principles for nonlinear elliptic finite element problems on surfaces with boundary

2018 ◽  
Vol 40 (2) ◽  
pp. 1241-1265 ◽  
Author(s):  
János Karátson ◽  
Balázs Kovács ◽  
Sergey Korotov
Keyword(s):  
Finite Element ◽  
Maximum Principle ◽  
Elliptic Equations ◽  
Real Life ◽  
Maximum Principles ◽  
The Maximum Principle ◽  
Nonlinear Elliptic ◽  
Discrete Analogues ◽  
Second Order Elliptic Equations ◽  
The Given

AbstractThe maximum principle forms an important qualitative property of second-order elliptic equations; therefore, its discrete analogues, the so-called discrete maximum principles (DMPs), have drawn much attention owing to their role in reinforcing the qualitative reliability of the given numerical scheme. In this paper DMPs are established for nonlinear finite element problems on surfaces with boundary, corresponding to the classical pointwise maximum principles on Riemannian manifolds in the spirit of Pucci & Serrin (2007, The Maximum Principle. Springer). Various real-life examples illustrate the scope of the results.

Some local maximum principles along Ricci flows

2020 ◽  
pp. 1-20
Author(s):  
Man-Chun Lee ◽  
Luen-Fai Tam
Keyword(s):  
Maximum Principle ◽  
Ricci Flow ◽  
Direct Proof ◽  
Local Maximum ◽  
Maximum Principles ◽  
Kernel Estimates ◽  
Bounded Curvature ◽  
Noncompact Manifolds ◽  
Ricci Flows ◽  
Dirichlet Heat Kernel

Abstract In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

scholarly journals Extended weak maximum principles for parabolic partial differential inequalities on unbounded domains

2014 ◽  
Vol 470 (2167) ◽  
pp. 20140079 ◽  
Author(s):  
J. C. Meyer ◽  
D. J. Needham
Keyword(s):  
Maximum Principle ◽  
Unbounded Domain ◽  
Differential Inequality ◽  
Unbounded Domains ◽  
Comparison Theorems ◽  
Weight Functions ◽  
Maximum Principles ◽  
Differential Inequalities ◽  
Partial Differential ◽  
Conditional Maximum

In this paper, we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains, where the solutions satisfy a variety of growth/decay conditions on the unbounded domain. We establish a conditional maximum principle, which states that a solution u to a linear parabolic partial differential inequality satisfies a maximum principle whenever a suitable weight function can be exhibited. Our extended maximum principles are then established by exhibiting suitable weight functions and applying the conditional maximum principle. In addition, we include several specific examples, to highlight the importance of certain generic conditions, which are required in the statements of maximum principles of this type. Furthermore, we demonstrate how to obtain associated comparison theorems from our extended maximum principles.

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

scholarly journals On the stokes operator in general unbounded domains

2009 ◽  
Vol 38 (1) ◽  
pp. 111-136 ◽  
Author(s):  
Reinhard FARWIG ◽  
Hideo KOZONO ◽  
Hermann SOHR
Keyword(s):  
Unbounded Domains ◽  
Stokes Operator ◽  
General Unbounded Domains

scholarly journals Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lingling Zhang ◽  
Hui Wang
Keyword(s):  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Gradient Term ◽  
Nonlinear Parabolic Problems ◽  
Nonlinear Parabolic ◽  
Robin Boundary ◽  
Upper Estimate

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:(b(u))t=∇·(h(t)k(x)a(u)∇u)+f(x,u,|∇u|2,t), inD×(0,T),(∂u/∂n)+γu=0, on∂D×(0,T),u(x,0)=u0(x)>0, inD¯, whereD⊂RN  (N≥2)is a bounded domain with smooth boundary∂D. Under some appropriate assumption on the functionsf,h,k,b, andaand initial valueu0, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

Local maximum principle satisfying high-order non-oscillatory schemes

2015 ◽  
Vol 81 (11) ◽  
pp. 689-715 ◽  
Author(s):  
Ritesh Kumar Dubey ◽  
Biswarup Biswas ◽  
Vikas Gupta
Keyword(s):  
Maximum Principle ◽  
Local Maximum ◽  
High Order
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