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Fundamental matrices and Green matrices for non-homogeneous elliptic systems

Blair Davey, Jonathan Hill, and Svitlana Mayboroda

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In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi–Nash–Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schrödinger ($-\Delta+V$) and generalized Schrödinger ($-\operatorname{div} A\nabla +V$) operators with real and complex coefficients, on arbitrary domains.

Article information

Publ. Mat., Volume 62, Number 2 (2018), 537-614.

Received: 9 January 2017
Revised: 23 April 2017
First available in Project Euclid: 16 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A08: Fundamental solutions 35J08: Green's functions 35B45: A priori estimates 35J107 35J57: Boundary value problems for second-order elliptic systems

Fundamental solution Green function elliptic equations Schrödinger operator


Davey, Blair; Hill, Jonathan; Mayboroda, Svitlana. Fundamental matrices and Green matrices for non-homogeneous elliptic systems. Publ. Mat. 62 (2018), no. 2, 537--614. doi:10.5565/PUBLMAT6221807.

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