Question

Prove the following Green's identity for function.....

4. (a) Prove the following Green's identity for functions f.g E Co(2) where2C R'" where the notation : ▽ Vf n, where n is the outward pointing unit normal vector. You may use the divergence theorem, as well as the identity (b) Let G(x.xo) denote the Green's function for the Laplacian on Ω with Dirichlet boundary con- ditions, that is, 4,G(x, xo) = δ(x-xo), for x 62 (x,x;)= 0 for x Eon By using part (a) and considering the difference show that G(X1, xo) = G(x0.X1) for any Xo, X1 62

Answer 1

ast R.H.S n ds 3

R.H.S