scholarly journals Asymptotic representation of the spectral function of self-adjoint elliptic operators of the second order with variable coefficients

1960 ◽  
Vol 1 (3-4) ◽  
pp. 423-483 ◽  
Author(s):  
F.J Bureau
Keyword(s):  
Spectral Function ◽  
Asymptotic Representation ◽  
Variable Coefficients ◽  
Elliptic Operators ◽  
Second Order

scholarly journals Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

2021 ◽  
Author(s):  
Hans-Christoph Grunau
Keyword(s):  
Principal Part ◽  
Present Note ◽  
Variable Coefficients ◽  
Higher Order ◽  
Elliptic Operators ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Green Functions ◽  
Polyharmonic Operator ◽  
Bounded Smooth

AbstractEstimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order $$2m\ge 4$$ 2 m ≥ 4 have been shown so far only when the principal part of L is the polyharmonic operator $$(-\Delta )^m$$ ( - Δ ) m . In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.

scholarly journals Heat Kernels of Second Order Complex Elliptic Operators and Applications

1998 ◽  
Vol 152 (1) ◽  
pp. 22-73 ◽  
Author(s):  
Pascal Auscher ◽  
Alan McIntosh ◽  
Philippe Tchamitchian
Keyword(s):  
Elliptic Operators ◽  
Heat Kernels ◽  
Second Order ◽  
Order Complex
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