On a self-adjoint operator associated with elliptic problems with a spectral parameter in the boundary conditions

1973 ◽  
Vol 24 (6) ◽  
pp. 682-686
Author(s):  
L. I. Tyutyunnik
Keyword(s):  
Boundary Conditions ◽  
Spectral Parameter ◽  
Adjoint Operator ◽  
Elliptic Problems

Elliptic problems for a Douglis–Nirenberg system over ℝn and over an exterior subregion

2016 ◽  
Vol 146 (3) ◽  
pp. 579-594
Author(s):  
M. Faierman
Keyword(s):  
Sobolev Space ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Complex Plane ◽  
Spectral Parameter ◽  
Differential Operators ◽  
Elliptic Problems ◽  
Fredholm Theory ◽  
Space Setting

We consider both a problem over ℝn and a boundary problem over an exterior subregion for a Douglis–Nirenberg system of differential operators under limited smoothness assumptions as well as under the assumption of parameter ellipticity in a closed sector Ꮭ in the complex plane with vertex at the origin. We pose each problem on an Lp Sobolev space setting, 1 < p < ∞, and denote by the operator induced in this setting by the problem over ℝn and by Ap the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for each of these operators for values of the spectral parameter λ lying in Ꮭ as well as results for these values of λ pertaining to the invariance of their Fredholm domains with p.

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