Fundamental matrices and Green matrices for non-homogeneous elliptic systems. (English) Zbl 1400.35098

Summary: 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 (\(-\mathrm{div} A\nabla +V\)) operators with real and complex coefficients, on arbitrary domains.


35J57 Boundary value problems for second-order elliptic systems
35J08 Green’s functions for elliptic equations
35A08 Fundamental solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
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