Let $B$ be the open unit ball in $\mathbb{R}^2$ and $r=\sqrt{x^2+y^2}$, consider $$ \begin{cases} \Delta u = c \qquad \text{in} \ B \\ \frac{\partial u}{\partial r} =0 \qquad \text{on} \ \partial B \end{cases} $$ For which values of $c\in \mathbb{R}$ does a solution exist? Is $c=0$ the only possibility?
2026-03-29 13:22:14.1774790534
Constant Laplacian with some boundary data
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