Let B(0, 2) = {(x, y) ∈ ℝ2 : x2 + y2 < 4}, and ∂B denote the boundary of B(0, 2). Assume (α, β) ≠ (0, 0), k ∈ ℝ, and u is any solution to
\(\rm \left\{\begin{matrix}-\Delta u=0& \rm in\:\:B(0, 2)\\\ \alpha u(x, y)+β\frac{∂ u}{∂ \nu}(x, y)=1+(x^2+y^2)k&\rm on\ ∂ B\end{matrix}\right.\)
where v(x, y) is the unit outward normal to B(0, 2) at (x, y) ∈ ∂B. Consider the following statements:
S1 : If β = 0 then there exists a (x0, y0) € B(0, 2) such that |u(x0, y0)| = \(\rm \frac{|1+4k|}{|α|}\)
S2 : If α = 0, then k = \(-\frac{1}{4}\)
Then
1
S1 is true but S2 is false
2
S2 is true but S1 is false
3
both S1 and S2 are true
4
both S1 and S2 are false