4. Extra Credit. (10 Points.) Consider the problem of

Last updated: 8/12/2022

4. Extra Credit. (10 Points.) Consider the problem of solving for functions u(x, y), defined on x² + y² ≤ 1, such that (i) Au = 0, and (ii) u = 0 at all points in the boundary x² + y² = 1. Show that the only solution to this problem is the zero function: u(x, y) = 0. Hint: Consider the vector field F = uvu. Can you make the same conclusion if we replace replace condition (ii) with "Vun = 0 at all points on the boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusion about u you can make? Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen- tial equations.