The poisson equation with weights in exterior domains of Rⁿ

Abstract

We consider Poisson's equation in G with Dirichlet boundary condition u=Φ on ∂G. Here is an exterior domain with a smooth compact boundary [∂G]. In the present paper we prove existence and uniqueness of the solution u in the Soboley space parovided . Here the aditional condition f ε L^ρ(G) is necessary and corresponds to a certain decay behaviour of f. We show that u tends to zero as |x| → ∞ if r > n. This result can be applied to weighted versions of Poisson's equation and leads to statements about the rate of decay of the solution

Keywords:

• Poisson equation
• exterior domain
• weights

AMS(MOS)::

• 35B40
• 35J05
• 35J25