ANTI-MAXIMUM PRINCIPLES FOR INDEFINITE-WEIGHT ELLIPTIC PROBLEMS

Anti-maximum Principles for Indefinite-weight Elliptic Problems

λ0 = λ0(1, P, Ω) := sup{λ ∈ R : ∃u > 0 s.t. (P − λ)u ≥ 0 in Ω}, where 1 is the constant function on Ω, taking at any point x ∈ Ω the value 1. Using the Krein-Rutman theorem, the author proved in [13, 14] generalized maximum principles and anti-maximum principles (in brief, GMPs and AMPs, respectively) for the problem (P − λ)uλ = f 0 in Ω. (1.1) In particular, these GMPs and AMPs (without weight...

On discrete maximum principles for nonlinear elliptic problems

Domain Decomposition Algorithms for Indefinite Elliptic Problems

Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane. We first consider an additive Schwarz...

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

New Maximum Principles for Linear Elliptic Equations

We prove extensions of the estimates of Aleksandrov and Bakel′man for linear elliptic operators in Euclidean space R to inhomogeneous terms in L spaces for q < n. Our estimates depend on restrictions on the ellipticity of the operators determined by certain subcones of the positive cone. We also consider some applications to local pointwise and L estimates.