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Abstract view

# Existence of Solutions to Poisson's Equation

Let $\Omega$ be a domain in $\mathbb R^n$ ($n\geq 2$). We find a necessary and sufficient topological condition on $\Omega$ such that, for any measure $\mu$ on $\mathbb R^n$, there is a function $u$ with specified boundary conditions that satisfies the Poisson equation $\Delta u=\mu$ on $\Omega$ in the sense of distributions.