Laplace's Equation — Question 4

Question 4 – Solution

Strategy. The Poisson kernel supplies a bounded harmonic extension without imposing impossible continuity at jumps.

Step 1: Compute and damp the Fourier modes. The average is $1/2$. Symmetry makes every sine coefficient zero, while $$a_n=\frac 1\pi\int_{-\pi/2}^{\pi/2}\cos(n\phi)\,d\phi =\frac{2\sin(n\pi/2)}{n\pi}.$$ The kernel expansion therefore gives $$\boxed{u(r,\theta)=\frac 12+\frac 2\pi\sum_{k\ge 0} \frac{(-1)^k r^{2k+1}\cos((2k+1)\theta)}{2k+1}.}$$ On every smaller closed disk all derivatives converge uniformly by geometric decay. Each polar harmonic is the real part of $(x+iy)^{2k+1}$.

Step 2: Use positivity and specify uniqueness. Both semicircles have positive length, and the kernel is strictly positive inside. Thus their harmonic weights lie strictly between zero and one, giving $0<u<1$ and $u(0)=1/2$. Uniqueness holds among bounded harmonic functions with these radial traces almost everywhere. Indeed, for the difference $w$, the Poisson formula on a circle of radius $R<1$ expresses a fixed interior value through $w(R,\phi)$. As $R\uparrow 1$, boundedness and the a.e. zero trace let dominated convergence give $w=0$. No common continuous trace at the jumps is required.

Step 3: Distinguish the boundary limits. At a point strictly within either semicircle the kernel is an approximate identity: away from any fixed angular neighborhood it tends uniformly to zero, while its total mass is fixed. The radial limit is therefore the local constant, one or zero. At $\theta=\pm\pi/2$ every displayed cosine term vanishes, so $u(r,\pm\pi/2)=1/2$ for all $r<1$. Neighboring boundary limits disagree; assigning $1/2$ at the jumps cannot make the closed-disk extension continuous.

Step 4: Sum the diameter profile. For $-1<x<1$, integrate the geometric series for $1/(1+x^2)$ from zero to $x$ to obtain $\arctan x=\sum_{k\ge 0}(-1)^kx^{2k+1}/(2k+1)$. Hence $$\boxed{u(x,0)=\tfrac 12+\tfrac 2\pi\arctan x.}$$ Its derivative is $2/[\pi(1+x^2)]>0$; its limits at $x=-1,1$ are zero and one. These endpoints lie on the open constant-data arcs, not at the jumps.