scholarly journals Eigenva1ue Distribution of Invariant Linear Second Order Elliptic Differential Operators with Constant Coefficients

1993 ◽  
Vol 12 (2) ◽  
pp. 285-296
Author(s):  
M. Belger
Keyword(s):  
Differential Operators ◽  
Second Order ◽  
Constant Coefficients ◽  
Elliptic Differential Operators

scholarly journals Unique continuation property of solutions to general second order elliptic systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Naofumi Honda ◽  
Ching-Lung Lin ◽  
Gen Nakamura ◽  
Satoshi Sasayama
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Unique Continuation ◽  
Elliptic Systems ◽  
General System ◽  
Second Order ◽  
Carleman Estimate ◽  
Unique Continuation Property ◽  
Constant Coefficients ◽  
Continuation Property

Abstract This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.

Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum

1988 ◽  
Vol 109 (1-2) ◽  
pp. 127-144 ◽  
Author(s):  
F. Fiedler
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Oscillation Criteria ◽  
Elliptic Differential Equations ◽  
Ordinary Differential Operators ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Lower Spectrum ◽  
Liouville Operators

SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

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