Poisson Equation

Assume that \(u\in C^2(\mathbb {R}^n\times [0,\infty ])\) satisfies the initial value problem for the wave equation \[ \begin {cases} u_{tt}-\Delta u=0, & \text {in }\mathbb {R}^n\times (0,\infty ),\\ u=g,\ u_t=h, & \text {on }\mathbb {R}^n\times \{t=0\}. \end {cases} \] Recall that when \(n=3\), using spherical means and the Euler-Poisson-Darboux equation, one can derive Kirchhoff’s formula \[ \begin {aligned} u(x,t) &= \frac {\partial }{\partial t}\left (t\fint _{\partial B(x,t)}g\dd S\right ) +t\fint _{\partial B(x,t)}h\dd S\\ &= \fint _{\partial B(x,t)} \bigl (th(y)+g(y)+Dg(y)\cdot (y-x)\bigr )\dd S(y),\qquad t>0. \end {aligned} \]

1. Outline the procedure (the “method of descent”) for deriving a similar formula (Poisson’s formula) for the initial value problem in the case \(n=2\).
2. Explain how a disturbance originating at a point \(x\in \mathbb {R}^2\) at time \(t=0\) propagates in relation to the cone \(C=\{(y,t):t>0,\ \lvert x-y\rvert <t\}\). In particular, using the Kirchhoff and Poisson formulas, show that there is a qualitative difference in propagation behavior between the cases \(n=2\) and \(n=3\).

Prove by means of the Poisson Formula that harmonic functions are of class \(C^\infty \).

Find the Green’s function for the Poisson equation \(\Delta u=f\) in the infinite strip \(0\le y\le 1\), \(-\infty <x<\infty \) with boundary data \(u(x,0)=0\) and \(u(x,1)=1\). Then solve the problem (in quadrature) if \(f(x,y)=y(1-y)e^{-x^2}\).