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

Sakellaris, Georgios

doi:10.2140/apde.2021.14.251

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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

DOI: 10.2140/apde.2021.14.251

Abstract

We construct Green’s functions for elliptic operators of the form $ℒ u = - div (A \nabla u + b u) + c \nabla u + d u$ in domains $Ω \subseteq ℝ^{n}$ , under the assumption $d \geq div b$ or $d \geq div c$ . We show that, in the setting of Lorentz spaces, the assumption $b - c \in L^{n, 1} (Ω)$ 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 $Ω \subseteq ℝ^{n}$ . 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 $ℒ u \leq - div f + g$ in the case $d \geq div c$ .

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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

Vol. 14, No. 1, 2021