0

Consider the Dirichlet boundary value problem for the Poisson-equation $$ -\Delta u=f\text{ in }B_R(0)\subset\mathbb{R}^3,~~~~~~~~~~u=0\text{ on }S_R(0) $$ with $f\in L^{\infty}(B_R(0))\cap C^{0,\lambda}(B_R(0))$ and $0<\lambda<1$.

(a) Give the solution $u\in C^{2,\lambda}(B_R(0))\cap C(\overline{B_R(0)})$ by use of the Green function $G(x,y)$ of $B_R(0)$.

(b) Give $u(0)$ for the case that $f(x)=f(\lVert x\rVert)$.

(c) For $\lVert x\rVert >1$ bring $u(x)$ in a form that can be written with the volume potential.

(d) Calculate the solution for the case $f\equiv 1, R=1$. (I can use Volume potential (show continuity))

Hello,

here Two little tasks concerning Dirichlet boundary value problem

I already solved (a) and (b).

Now I have to solve (c) and (d).

For (c) I do not know what is meant.

I already have

$$ u(x)=\frac{1}{4\pi}(U_3(x)-\int_{B_R(0)}\frac{f(y)}{\left\lVert \frac{\lVert y\rVert}{R}x-\frac{R}{\lVert y\rVert}y\right\rVert}\, dy), $$ but do not know if this is what is asked or not... maybe one can write the integral as a volume potential, too?

Community
  • 1
  • I think that's what you already have. Maybe in (a) he wanted a volume potential and a surface potential; that's an alternative representation. Using that method, you would use U3, and then create a Poisson integral P3 to match the boundary values of U3, leading to the same solution U3-P3 inside the ball because P3 is a solution of Laplace's equation inside the ball. Those two representations are equivalently inside the ball |x| < R, but definitely look different outside the ball. – Disintegrating By Parts Dec 23 '13 at 21:54
  • Sorry, I do not understand that. –  Jan 03 '14 at 15:09

0 Answers0