Question

problem Let $\Omega = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 < 4\}$ be the domain. Consider the Neumann $\begin{cases} \Delta u = 0, \quad (x, y) \in \Omega, \\ \frac{\partial u}{\partial \nu} = \alpha x^2 + \beta y + \gamma, \quad (x, y) \in \partial \Omega, \end{cases}$ where $\alpha$, $\beta$ and $\gamma$ are real constants. (a) Find the values of $\alpha$, $\beta$, $\gamma$ for which the problem is not solvable. (b) Solve the problem for those values of $\alpha$, $\beta$, $\gamma$ for which a solution does exist.

          problem
Let $\Omega = \{(x, y) \in \mathbb{R}^2 | x^2 + y^2 < 4\}$ be the domain. Consider the Neumann
$\begin{cases}
\Delta u = 0, \quad (x, y) \in \Omega, \\
\frac{\partial u}{\partial \nu} = \alpha x^2 + \beta y + \gamma, \quad (x, y) \in \partial \Omega,
\end{cases}$
where $\alpha$, $\beta$ and $\gamma$ are real constants.
(a) Find the values of $\alpha$, $\beta$, $\gamma$ for which the problem is not solvable.
(b) Solve the problem for those values of $\alpha$, $\beta$, $\gamma$ for which a solution does exist.

$problem Let Ω = {(x, y) ∈ℝ^2 | x^2 + y^2 < 4} be the domain. Consider the Neumann Δ u = 0, (x, y) ∈Ω, (∂ u)/(∂ν) = α x^2 + β y + γ, (x, y) ∈∂Ω, where α, β and γ are real constants. (a) Find the values of α, β, γ for which the problem is not solvable. (b) Solve the problem for those values of α, β, γ for which a solution does exist.$

Added by Kenneth S.

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