L2 estimates for commutators of the Dirichlet-to-Neumann Map associated to elliptic operators with complex-valued bounded measurable coefficients on R+n+1

2021 ◽  
pp. 125408
Author(s):  
Steve Hofmann ◽  
Guoming Zhang
Keyword(s):  
Elliptic Operators ◽  
Measurable Coefficients ◽  
Dirichlet To Neumann Map ◽  
Complex Valued

On the spectra of non-self-adjoint realisations of second-order elliptic operators

1981 ◽  
Vol 90 (1-2) ◽  
pp. 71-105 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Spherical Shell ◽  
Essential Spectrum ◽  
Weighted Space ◽  
Elliptic Operators ◽  
Second Order ◽  
Sectorial Operator ◽  
Image Position ◽  
Second Order Elliptic Operators ◽  
Complex Valued

SynopsisLet τ denote the second-order elliptic expressionwhere the coefficients bj and q are complex-valued, and let Ω be a spherical shell Ω = {x:x ∈ ℝn, l <|x|<m} with l≧0, m≦∞. Under the conditions assumed on the coefficients of τ and with either Dirichlet or Neumann conditions on the boundary of Ω, τ generates a quasi-m-sectorial operator T in the weighted space L2(Ω;w). The main objective is to locate the spectrum and essential spectrum of T. Best possible results are obtained.

scholarly journals Conical square functions for degenerate elliptic operators

2020 ◽  
Vol 13 (1) ◽  
pp. 75-113 ◽  
Author(s):  
Li Chen ◽  
José María Martell ◽  
Cruz Prisuelos-Arribas
Keyword(s):  
Divergence Form ◽  
Elliptic Operators ◽  
Square Functions ◽  
Degenerate Elliptic Operators ◽  
Weighted Estimates ◽  
Poisson Semigroup ◽  
Bounded Complex ◽  
Degenerate Elliptic ◽  
Complex Valued ◽  
Kato Square Root Problem

AbstractThe aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)}, where {w\in A_{2}} and A is an {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the {L^{2}(w)}-Kato square root problem obtaining that {\sqrt{L_{w}}} is equivalent to the gradient on {L^{2}(w)}. The same authors in collaboration with the second named author of this paper studied the {L^{p}(w)}-boundedness of operators that are naturally associated with {L_{w}}, such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in {L^{p}(v\,dw)} for {v\in A_{\infty}(w)}), and in particular a class of “degeneracy” weights w was found in such a way that the classical {L^{2}}-Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on {L^{p}(w)} and on {L^{p}(v\,dw)}, with {v\in A_{\infty}(w)}, of the conical square functions that one can construct using the heat or Poisson semigroup associated with {L_{w}}. As a consequence of our methods, we find a class of degeneracy weights w for which {L^{2}}-estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with {L_{w}}.

scholarly journals Spectral theory of elliptic differential operators with indefinite weights

2013 ◽  
Vol 143 (1) ◽  
pp. 21-38 ◽  
Author(s):  
Jussi Behrndt
Keyword(s):  
Differential Operators ◽  
Elliptic Operators ◽  
Weight Functions ◽  
Real Spectrum ◽  
Indefinite Weights ◽  
Elliptic Differential Operators ◽  
Indefinite Weight Functions ◽  
Dirichlet To Neumann Map ◽  
Second Order Elliptic Operators ◽  
Special Case

The spectral properties of a class of non-self-adjoint second-order elliptic operators with indefinite weight functions on unbounded domains Ω are investigated. It is shown, under an abstract regularity assumption, that the non-real spectrum of the associated elliptic operators in L2(Ω) is bounded. In the special case where Ω = ℝn decomposes into subdomains Ω+ and Ω− with smooth compact boundaries and the weight function is positive on Ω+ and negative on Ω−, it turns out that the non-real spectrum consists only of normal eigenvalues that can be characterized with a Dirichlet-to-Neumann map.

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