On the strong maximum principle for a fractional Laplacian

2021 ◽  
Author(s):  
Nguyen Ngoc Trong ◽  
Do Duc Tan ◽  
Bui Le Trong Thanh
Keyword(s):  
Maximum Principle ◽  
Fractional Laplacian ◽  
Strong Maximum Principle

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.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

scholarly journals The strong maximum principle revisited

2004 ◽  
Vol 196 (1) ◽  
pp. 1-66 ◽  
Author(s):  
Patrizia Pucci ◽  
James Serrin
Keyword(s):  
Maximum Principle ◽  
Strong Maximum Principle

scholarly journals A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients

2019 ◽  
Vol 21 (07) ◽  
pp. 1850057 ◽  
Author(s):  
Francesca Anceschi ◽  
Michela Eleuteri ◽  
Sergio Polidoro
Keyword(s):  
Maximum Principle ◽  
Harnack Inequality ◽  
Weak Solutions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Strong Maximum Principle ◽  
Kolmogorov Equation ◽  
Measurable Coefficients ◽  
Rough Coefficients ◽  
Vector Formula

We consider weak solutions of second-order partial differential equations of Kolmogorov–Fokker–Planck-type with measurable coefficients in the form [Formula: see text] where [Formula: see text] is a symmetric uniformly positive definite matrix with bounded measurable coefficients; [Formula: see text] and the components of the vector [Formula: see text] are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by Golse et al. As a corollary, we obtain a strong maximum principle.

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