Vol. 14, No. 1, 2021

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On scale-invariant bounds for the Green's function for second-order elliptic equations with lower-order coefficients and applications

Georgios Sakellaris

Vol. 14 (2021), No. 1, 251–299
Abstract

We construct Green’s functions for elliptic operators of the form u = div(Au + bu) + cu + du in domains Ω n , under the assumption d divb or d divc. We show that, in the setting of Lorentz spaces, the assumption b c Ln,1(Ω) is both necessary and optimal to obtain pointwise bounds for Green’s functions. We also show weak-type bounds for the Green’s function and its gradients. Our estimates are scale-invariant and hold for general domains Ω n . Moreover, there is no smallness assumption on the norms of the lower-order coefficients. As applications we obtain scale-invariant global and local boundedness estimates for subsolutions to u divf + g in the case d divc.

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Keywords
Green's function, fundamental solution, lower-order coefficients, pointwise bounds, Lorentz bounds, maximum principle, Moser-type estimate
Mathematical Subject Classification 2010
Primary: 35A08, 35J08, 35J15
Secondary: 35B50, 35J20, 35J86
Milestones
Received: 10 April 2019
Revised: 26 July 2019
Accepted: 26 September 2019
Published: 19 February 2021
Authors
Georgios Sakellaris
Department of Mathematics
Universitat Autònoma de Barcelona
Barcelona
Spain