Higher-Order Elliptic Equations with Constant Coefficients

1982 ◽  
pp. 185-205
Author(s):  
Fritz John
Keyword(s):  
Elliptic Equations ◽  
Higher Order ◽  
Constant Coefficients

scholarly journals Completeness Theorems for Elliptic Equations of Higher Order with Constant Coefficients

2007 ◽  
Vol 14 (1) ◽  
pp. 81-97
Author(s):  
Alberto Cialdea
Keyword(s):  
Elliptic Equation ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Complete System ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Polynomial Solutions ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
Completeness Theorems

Abstract Let {ω𝑘 } be a complete system of polynomial solutions of the elliptic equation ∑|α|⩽2𝑚 aα 𝐷 α 𝑢 = 0, aα being real constants. We give necessary and sufficient conditions for the completeness of the system in [𝐿𝑝(∂Ω)]𝑚, where Ω ⊂ is a bounded domain such that is connected and ∂Ω ∈ 𝐶1.

COMPLETENESS THEOREMS IN THE UNIFORM NORM CONNECTED TO ELLIPTIC EQUATIONS OF HIGHER ORDER WITH CONSTANT COEFFICIENTS

2012 ◽  
Vol 10 (01) ◽  
pp. 1-20 ◽  
Author(s):  
A. CIALDEA
Keyword(s):  
Differential Operator ◽  
Elliptic Equation ◽  
Bounded Domain ◽  
Characteristic Polynomial ◽  
Elliptic Equations ◽  
Uniform Norm ◽  
Normal Derivative ◽  
Higher Order ◽  
Constant Coefficients ◽  
Completeness Theorems

We consider the problem of the completeness of the system [Formula: see text] in [C0(Σ)]m, where {ωk} is a basis of polynomial solutions of the elliptic equation ∑|α|≤2m aαDαu = 0, aα are real constants, Σ is the boundary of a bounded domain in ℝn and ∂ν denotes the normal derivative. If E satisfies a Gårding inequality and [Formula: see text] is connected, we show that such a completeness property holds if and only if all the irreducible factors of the characteristic polynomial of the differential operator vanish at the origin. The proof hinges on some jump formulas obtained for general potentials generated by measures.

scholarly journals EIGENVALUE ESTIMATES FOR HIGHER ORDER ELLIPTIC EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 267-278
Author(s):  
HSU-TUNG KU ◽  
MEI-CHIN KU ◽  
XIN-MIN ZHANG
Keyword(s):  
Lower Bound ◽  
Elliptic Equations ◽  
Eigenvalue Problems ◽  
Higher Order ◽  
Bounded Domains ◽  
Dirichlet Eigenvalue ◽  
Eigenvalue Estimates

In this paper, we obtain good lower bound estimates of eigenvalues for various Dirichlet eigenvalue problems of higher order elliptic equations on bounded domains in $\mathbb{R}^n$.

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