Hi, got a couple of questions on my latest worksheet (non-assessed) that I'm struggling with. Any help would be appreciated.
1) Find an example of a function u: R^n -> R (where n > 2) that is twice differentiable everywhere, is subharmonic (Laplacian of u is greater than or equal to 0), is bounded, but NOT constant.
Finding this harder than expected. My functions usually fail to be twice differentiable at certain points (normally zero). For example, I tried defining u(x) =|x| for |x| < 1 and u(x) = 1 whenever |x| >= 1 but this fails to be differentiable at the origin, right?
2) Solve the following Dirichlet problem on the open ball of radius 2 centred at 0 (now denoted B2) in R^2.
Laplacian(u) = 1 in B2
u(x,y) = x^2 - y^2 - 3y + 4 on the boundary of B2
From my lecture notes it seems I am supposed to use Green's Function to solve this. I am still not sure how to work out the Greens Function for B2 (in class we have done it for the unit ball in dimension bigger than 2 but not for dimension 2). How do I do this? Or is there (I hope) another way to solve this?