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 Rn 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 (-Δ+V ) and generalized Schrödinger (-divA∇+V ) operators with real and complex coefficients, on arbitrary domains.
Bibliographical noteFunding Information:
Davey is supported in part by the Simons Foundation [Grant number 430198]. Mayboroda is supported in part by the Alfred P. Sloan Fellowship; the NSF INSPIRE Award [DMS 1344235]; NSF CAREER Award [DMS 1220089]; NSF UMN MRSEC Seed grant [DMR 0212302].
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- Elliptic equations
- Fundamental solution
- Green function
- Schrödinger operator