A maximum principle for circle-valued temperatures

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrew A. Cooper
Keyword(s):  
Maximum Principle ◽  
Heat Equation ◽  
Riemannian Manifolds ◽  
Time Dependent

Abstract In this note we prove a ‘maximum principle’ for circle-valued solutions of the (time-dependent) heat equation on closed Riemannian manifolds.

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 A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

1998 ◽  
Vol 151 ◽  
pp. 25-36 ◽  
Author(s):  
Kensho Takegoshi
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Volume Growth ◽  
Complete Riemannian Manifold ◽  
Volume Estimate ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Abstract.A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

Some New Results on L2 Cohomology of Negatively Curved Riemannian Manifolds

2005 ◽  
Vol 57 (2) ◽  
pp. 251-266
Author(s):  
M. Cocos
Keyword(s):  
Heat Equation ◽  
Riemannian Manifolds ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Regularity Properties ◽  
L2 Cohomology ◽  
Cohomology Spaces

AbstractThe present paper is concerned with the study of the L2 cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial L2 cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms.

scholarly journals hp-FEM for the fractional heat equation

2020 ◽  
Author(s):  
Jens Markus Melenk ◽  
Alexander Rieder
Keyword(s):  
Finite Elements ◽  
Heat Equation ◽  
Exponential Convergence ◽  
Time Dependent ◽  
Nonlocal Operator ◽  
Time Stepping ◽  
Fractional Heat Equation ◽  
Time Dependent Problem ◽  
Hp Finite Elements ◽  
Abstract Framework

Abstract We consider a time-dependent problem generated by a nonlocal operator in space. Applying for the spatial discretization a scheme based on $hp$-finite elements and a Caffarelli–Silvestre extension we obtain a semidiscrete semigroup. The discretization in time is carried out by using $hp$-discontinuous Galerkin based time stepping. We prove exponential convergence for such a method in an abstract framework for the discretization in the spatial domain $\varOmega $.

scholarly journals Integral maximum principle and its applications

1994 ◽  
Vol 124 (2) ◽  
pp. 353-362 ◽  
Author(s):  
Alexander Grigor'yan
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Heat Equation ◽  
Heat Kernel ◽  
Integral Maximum Principle ◽  
Double Integrals

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

scholarly journals On the parabolic potentials in degenerate-type heat equation

1991 ◽  
Vol 4 (2) ◽  
pp. 147-160 ◽  
Author(s):  
Igor Malyshev
Keyword(s):  
Heat Equation ◽  
Mixed Type ◽  
Time Dependent ◽  
Degenerate Type ◽  
Theory Technique

Using distributions theory technique we introduce parabolic potentials for the heat equation with one time-dependent coefficient (not everywhere positive and continuous) at the highest space-derivative, discuss their properties, and apply obtained results to three illustrative problems. Presented technique allows to deal with some equation of the degenerate/mixed type.

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