Laplace's Equation — Question 9

Question 9 – Solution

Strategy. Harmonic continuation damps a mode by $\rho^N$; inversion must divide by exactly that small factor.

Step 1: Recover the two parameters. The regular harmonic extension is $u=A+Br^N\cos(N\theta)$. For the measured trace $h(\theta)=u(\rho,\theta)$, define $$m=\frac 1{2\pi}\int_0^{2\pi}h(\theta)\,d\theta,\qquad c_N=\frac 1\pi\int_0^{2\pi}h(\theta)\cos(N\theta)\,d\theta.$$ Orthogonality gives $$\boxed{A=m,\qquad B=\rho^{-N}c_N.}$$ Consistency with the stated model requires every sine coefficient and every nonconstant cosine coefficient except $c_N$ to vanish. Since $\rho>0$, the two parameters are unique.

Step 2: Reconstruct the given field. Here $\rho^N=1/64$, $m=2$, and $c_6=0.03=3/100$, so $B=48/25=1.92$. Thus $$\boxed{g(\theta)=2+\frac{48}{25}\cos(6\theta),\qquad u(r,\theta)=2+\frac{48}{25}r^6\cos(6\theta).}$$ Substitution at $r=1/2$ reproduces the measured coefficient $3/100$. The polar mode is a Cartesian harmonic polynomial and is regular at the center.

Step 3: Quantify inverse amplification. A coefficient error $e$, $|e|\le\eta$, produces error $\rho^{-N}e$ in $B$. Consequently $|\delta B|\le\eta\rho^{-N}$, with equality when $|e|=\eta$. The desired bound is met exactly when $$N\le\frac{\log(\varepsilon/\eta)}{\log(1/\rho)},\qquad \boxed{N_{\max}=\left\lfloor\frac{\log(\varepsilon/\eta)} {\log(1/\rho)}\right\rfloor.}$$ If this integer is zero, no positive mode qualifies. The given numbers give $N_{\max}=6$: mode six has error bound $0.0064$, whereas mode seven has $0.0128$. This is a per-coefficient criterion, not a bound on a sum of many mode errors.

Step 4: Separate forward and inverse stability. For continuous boundary perturbations, the maximum principle gives $\|u_1-u_2\|_{\infty,{\mathrm{disk}}}\le\|g_1-g_2\|_{\infty,\partial{\mathrm{disk}}}$. In contrast, take $g_n(\theta)=\cos(n\theta)$ and $u_n=r^n\cos(n\theta)$. Then $\|g_n\|_\infty=1$ but $\|u_n(\rho,\cdot)\|_\infty=\rho^n\to 0$. Thus no fixed bound can control boundary sup-norm errors by measured-circle sup-norm errors across all modes, although each exact finite-mode inversion is unique.