Construction and properties for the Green's function with Neumann boundary condition

2025/10/17 by Antoine Bricmont, Bricmont, Antoine
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2510.15544

openalex publication_date 2025/10/17 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28

Abstract

This article addresses the construction and analysis of the Green's function for the Neumann boundary value problem associated with the operator -Δ+ a on a smooth bounded domain Ω⊂ ℝ^N (N ≥ 3) with a∈ L^∞(Ω). Under the assumption that -Δ+ a is coercive, we obtain the existence, uniqueness, and qualitative properties of the Green's function G(x,y). The Green's function G(x,y) is constructed explicitly, satisfying pointwise estimates and derivative estimates near the singularity. Also, near the boundary of Ω, G is compared to the Green's function of the laplacian, with pointwise estimates. Other properties, like symmetry and positivity among other things, are established.

Citations

• Multiplicity and stability of the Pohozaev obstruction for\n Hardy-Schr "odinger equations with boundary singularity
• Examples of compact Einstein four-manifolds with negative curvature
• Stability and instability of the Einstein-Lichnerowicz constraint system
• The Lin-Ni's problem for mean convex domains

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