Starting with the fundamental solution for A in dimension N = 2 (cf. Definition 8.5.1), find...

 

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Starting with the fundamental solution for A in dimension N = 2 (cf. Definition 8.5.1), find the Green's function for the open disc of radius a: D = {(x, y)|x + y < a}. Use the Green's function to show that the solution to the Dirichlet problem in D, Au = 0 in D, u = g on D, is given by 1 a-xol g(x) dSx. u(xo) 2 S - x=a This is known as the Poisson formula in 2D. Here g is a continuous function on the boundary circle. Definition 8.5.1. The fundamental solution for A in dimension N (N 2) is defined to be 27 log || in dimension N = 2, 1 0(x) === - in dimension N = 3, 4|x| 1 in dimensions N > 3, N(N-2)WN|x|N-2 where wN is the (hyper)volume of the unit sphere in RN. Starting with the fundamental solution for A in dimension N = 2 (cf. Definition 8.5.1), find the Green's function for the open disc of radius a: D = {(x, y)|x + y < a}. Use the Green's function to show that the solution to the Dirichlet problem in D, Au = 0 in D, u = g on D, is given by 1 a-xol g(x) dSx. u(xo) 2 S - x=a This is known as the Poisson formula in 2D. Here g is a continuous function on the boundary circle. Definition 8.5.1. The fundamental solution for A in dimension N (N 2) is defined to be 27 log || in dimension N = 2, 1 0(x) === - in dimension N = 3, 4|x| 1 in dimensions N > 3, N(N-2)WN|x|N-2 where wN is the (hyper)volume of the unit sphere in RN.