Vector Calculus | Part IA, 2001

In R3\mathbb{R}^{3} show that, within a closed surface SS, there is at most one solution of Poisson's equation, 2ϕ=ρ\nabla^{2} \phi=\rho, satisfying the boundary condition on SS

αϕn+ϕ=γ\alpha \frac{\partial \phi}{\partial n}+\phi=\gamma \text {, }

where α\alpha and γ\gamma are functions of position on SS, and α\alpha is everywhere non-negative.

Show that

ϕ(x,y)=e±lxsinly\phi(x, y)=e^{\pm l x} \sin l y

are solutions of Laplace's equation 2ϕ=0\nabla^{2} \phi=0 on R2\mathbb{R}^{2}.

Find a solution ϕ(x,y)\phi(x, y) of Laplace's equation in the region 0<x<π,0<y<π0<x<\pi, 0<y<\pi that satisfies the boundary conditions

ϕ=0 on 0<x<πy=0ϕ=0 on 0<x<πy=πϕ+ϕ/n=0 on x=00<y<πϕ=sin(ky) on x=π0<y<π\begin{array}{cccc} \phi=0 & \text { on } & 0<x<\pi & y=0 \\ \phi=0 & \text { on } & 0<x<\pi & y=\pi \\ \phi+\partial \phi / \partial n=0 & \text { on } & x=0 & 0<y<\pi \\ \phi=\sin (k y) & \text { on } & x=\pi & 0<y<\pi \end{array}

where kk is a positive integer. Is your solution the only possible solution?

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