scholarly journals SCATTERING THEORY FOR ELLIPTIC DIFFERENTIAL OPERATORS IN UNBOUNDED DOMAINS.

1972 ◽  
Author(s):  
C.I. Goldstein
Keyword(s):  
Scattering Theory ◽  
Differential Operators ◽  
Unbounded Domains ◽  
Elliptic Differential Operators

Oscillation theorems for semilinear elliptic differential operators

1978 ◽  
Vol 82 (1-2) ◽  
pp. 135-151 ◽  
Author(s):  
Manabu Naito ◽  
Norio Yoshida
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Differential Inequalities ◽  
Elliptic Differential Operator ◽  
Semilinear Elliptic ◽  
All Solutions ◽  
Existence Of Positive Solutions ◽  
Elliptic Differential Operators ◽  
Oscillation Theorems

SynopsisThe semilinear elliptic differential operator L[u] = Δu + c(x, u) is studied and sufficient conditions are derived for all solutions of uL[u] ≦ 0 with suitable boundary conditions to be oscillatory in unbounded domains of Rn. Here, unbounded domains to be considered are cones, strips and cylinders in Rn. The results are based on the conditions for the non-existence of positive solutions of ordinary differential inequalities.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
Author(s):  
P. Drábek ◽  
A. Kufner ◽  
V. Mustonen
Keyword(s):  
Sobolev Spaces ◽  
Differential Operators ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Operators ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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