scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

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.

Saint-Venant’s Principle and the Torsion of Thin Shells of Revolution

1976 ◽  
Vol 43 (4) ◽  
pp. 663-667 ◽  
Author(s):  
C. O. Horgan ◽  
L. T. Wheeler
Keyword(s):  
Maximum Principle ◽  
Exponential Decay ◽  
Elliptic Equations ◽  
Second Order ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Quantitative Characterization ◽  
Elastic Shells ◽  
The Maximum Principle

This paper is concerned with obtaining stress estimates for the problem of axisymmetric torsion of thin elastic shells of revolution subject to self-equilibrated end loads. The results are obtained in the form of explicit pointwise stress bounds exhibiting an exponential decay with distance from the ends, thus supplying a quantitative characterization of Saint-Venant’s principle for this problem. In contrast to arguments using energy inequalities, here we apply a technique, recently developed by the authors, based on the maximum principle for second-order uniformly elliptic equations.

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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