Very weak solutions of Poisson’s equation with singular data under Neumann boundary conditions

• Published in: Calculus of variations and partial differential equations Vol. 52; no. 3-4; pp. 705 - 726
• Main Authors: Merker, Jochen, Rakotoson, Jean-Michel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2015; Springer Nature B.V
• Subjects: Analysis; Boundaries; Boundary conditions; Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Partial differential equations; Poisson distribution; Poisson equation; Regularity; Systems Theory; Texts; Theorems; Theoretical; Uniqueness theorems; 35D10; MSC 35J25
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-014-0730-0

In this article we consider the problem to find a very weak solution u ∈ L 1 ( Ω ) of Poisson’s equation - Δ u = f in a smooth bounded domain Ω ⊂ R N for a singular right hand side f under Neumann boundary conditions on ∂ Ω . We prove a general existence and uniqueness theorem and discuss regularity of very weak solutions. Particularly, we are able to generalize corresponding results for Poisson’s problem with f ∈ L 1 ( Ω ) and Neumann boundary condition ∂ u ∂ n = - ∫ Ω f ( x ) d x δ x 0 for a point x 0 ∈ ∂ Ω to non-integrable f with f ( x ) | x - x 0 | ∈ L 1 ( Ω ) . Applications to existence for singular data and properties of boundary Green functions are presented.