Elliptic and Parabolic Maximum Principles

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

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

Journal of Differential Equations ◽

10.1016/j.jde.2021.07.008 ◽

2021 ◽

Vol 298 ◽

pp. 346-386

Author(s):

Tomasz Klimsiak ◽

Tomasz Komorowski

Keyword(s):

Differential Operators ◽

Maximum Principles ◽

Quantitative Version

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On enforcing maximum principles and achieving element-wise species balance for advection–diffusion–reaction equations under the finite element method

Journal of Computational Physics ◽

10.1016/j.jcp.2015.09.057 ◽

2016 ◽

Vol 305 ◽

pp. 448-493 ◽

Cited By ~ 13

Author(s):

M.K. Mudunuru ◽

K.B. Nakshatrala

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Diffusion Reaction ◽

Maximum Principles ◽

The Finite Element Method ◽

Advection Diffusion ◽

Reaction Equations ◽

Advection Diffusion Reaction Equations ◽

Element Method

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Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

Abstract and Applied Analysis ◽

10.1155/2014/165429 ◽

2014 ◽

Vol 2014 ◽

pp. 1-28

Author(s):

Jiang Zhu ◽

Dongmei Liu

Keyword(s):

Time Scales ◽

Initial Value Problems ◽

Second Order ◽

Dynamic Equations ◽

Maximum Principles ◽

Uniqueness Theorems ◽

Initial Value ◽

Approximation Theorems ◽

Construction Techniques ◽

Linear And Nonlinear

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

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