scholarly journals On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

2021 ◽  
Vol 298 ◽  
pp. 346-386
Author(s):  
Tomasz Klimsiak ◽  
Tomasz Komorowski
Keyword(s):  
Differential Operators ◽  
Maximum Principles ◽  
Quantitative Version

scholarly journals MAXIMUM PRINCIPLES AROUND AN EIGENVALUE WITH CONSTANT EIGENFUNCTIONS

2008 ◽  
Vol 10 (06) ◽  
pp. 1243-1259 ◽  
Author(s):  
J. CAMPOS ◽  
J. MAWHIN ◽  
R. ORTEGA
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Neumann Boundary ◽  
Function Spaces ◽  
Strong Maximum Principle ◽  
Neumann Boundary Conditions ◽  
Linear Operators ◽  
Maximum Principles ◽  
Partial Differential Operators

A class of linear operators L + λI between suitable function spaces is considered, when 0 is an eigenvalue of L with constant eigenfunctions. It is proved that L + λI satisfies a strong maximum principle when λ belongs to a suitable pointed left-neighborhood of 0, and satisfies a strong uniform anti-maximum principle when λ belongs to a suitable pointed right-neighborhood of 0. Applications are given to various types of ordinary or partial differential operators with periodic or Neumann boundary conditions.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Sign in / Sign up

Export Citation Format

Share Document