Answer to Question #272006 in Differential Equations for may

Question

Let u(x,y) be the harmonic function in D = {(x,y)|x² + y² < 36} which satisfies the Dirichlet

boundary condition

u(x,y) = x , x<0

u(x,y) = 0 , otherwise

Prove that u(x,y) < min(x,0) in D.

Evaluate u(0,0) using the mean value principle.

Using Poisson’s formula evaluate u(0,y) for 0 ≤y < 6.

Using the method of separation of variables, find the solution u(x,y) in D