On a strong maximum principle for fully nonlinear subelliptic equations with Hörmander condition

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Tilak Bhattacharya ◽  
Ahmed Mohammed
Keyword(s):  
Maximum Principle ◽  
Strong Maximum Principle ◽  
Fully Nonlinear ◽  
Hörmander Condition ◽  
Subelliptic Equations

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 Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2985
Author(s):  
Georgi Boyadzhiev ◽  
Nikolai Kutev
Keyword(s):  
Maximum Principle ◽  
Viscosity Solutions ◽  
Principal Symbol ◽  
Elliptic Systems ◽  
Strong Maximum Principle ◽  
Nonlinear Elliptic Systems ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Weakly Coupled ◽  
First Time

In this paper, we consider the validity of the strong maximum principle for weakly coupled, degenerate and cooperative elliptic systems in a bounded domain. In particular, we are interested in the viscosity solutions of elliptic systems with fully nonlinear degenerated principal symbol. Applying the method of viscosity solutions, introduced by Crandall, Ishii and Lions in 1992, we prove the validity of strong interior and boundary maximum principle for semi-continuous viscosity sub- and super-solutions of such nonlinear systems. For the first time in the literature, the strong maximum principle is considered for viscosity solutions to nonlinear elliptic systems. As a consequence of the strong interior maximum principle, we derive comparison principle for viscosity sub- and super-solutions in case when on of them is a classical one. The main novelty of this work is the reduction of the smoothness of the solution. In the literature the strong maximum principle is proved for classical C2 or generalized C1 solutions, while we prove it for semi-continuous ones.

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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