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 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.

Nonlinear Elliptic Equations in Strip-Like Domains

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Jaeyoung Byeon ◽  
Kazunaga Tanaka
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

AbstractWe study the existence of a positive solution of a nonlinear elliptic equationwhere k ≥ 2 and D is a bounded domain domain in R

scholarly journals Fundamental Solution of Elliptic Equation with Positive Definite Matrix Coefficient

CAUCHY ◽  
2018 ◽  
Vol 5 (2) ◽  
pp. 64
Author(s):  
Khoirunisa Khoirunisa ◽  
Corina Karim
Keyword(s):  
Elliptic Equation ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elliptic Equations ◽  
Positive Definite Matrix ◽  
Radial Solution ◽  
Positive Definite ◽  
Real Constant ◽  
Constant Coefficients

<p>In this paper, we study the fundamental solution of elliptic equations with real constant coefficients  </p><p class="Body">where is a positive definite matrix. We obtained by searching the radial solution so that we solved the equation into ordinary differential equations.</p><h1> </h1>

Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane

2020 ◽  
Vol 17 (4) ◽  
pp. 594-600
Author(s):  
Olga Trofymenko
Keyword(s):  
Elliptic Equation ◽  
Complex Plane ◽  
Elliptic Equations ◽  
Regular Polygon ◽  
Mean Value ◽  
Linear Elliptic Equation ◽  
Mean Value Theorems ◽  
Polynomial Solutions ◽  
Constant Coefficients ◽  
The Mean

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.

Minimum action solutions of non-linear elliptic equations in unbounded domains containing a plane

1986 ◽  
Vol 102 (3-4) ◽  
pp. 327-343 ◽  
Author(s):  
Otared Kavian
Keyword(s):  
Elliptic Equation ◽  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Necessary And Sufficient Conditions ◽  
Smooth Bounded Domain ◽  
Non Linear ◽  
Necessary And Sufficient

SynopsisLet d ≧ 1 be an integer and ω ⊂ℝd a smooth bounded domain and consider the elliptic equation − Δu = g(u) on Ω = ℝ2 × ω. We prove that under (almost) necessary and sufficient conditions on the continuous function g: ℝm→ ℝm the above equation has a minimum-action solution.

Quasilinear Elliptic Equations with Singular Nonlinearity

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
João Marcos do Ó ◽  
Esteban da Silva
Keyword(s):  
Differential Operator ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Quasilinear Elliptic Equations ◽  
Mems Devices ◽  
Singular Nonlinearity ◽  
Electrostatic Mems ◽  
Quasilinear Elliptic

AbstractIn this paper, motivated by recent works on the study of the equations which model electrostatic MEMS devices, we study the quasilinear elliptic equationAccording to the choice of the parameters α, β, and γ, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the

scholarly journals Degenerate elliptic equations and nonuniqueness of solutions to the Kolmogorov equation

2019 ◽  
Vol 487 (4) ◽  
pp. 361-364
Author(s):  
T. I. Krasovitskii
Keyword(s):  
Elliptic Equation ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Degenerate Elliptic Equation ◽  
New Method ◽  
Kolmogorov Equation ◽  
Degenerate Elliptic Equations ◽  
Nonuniqueness Of Solutions ◽  
Degenerate Elliptic ◽  
Fokker Planck

In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker-Planck-Kolmogorov equation to a degenerate elliptic equation on a bounded domain.

Sign in / Sign up

Export Citation Format

Share Document