Terminology — Question 9

Question 9 – Solution

Strategy. Small absolute outputs are not enough for stability; compare their size with the much smaller input, then normalize.

Step 1: Verify the supplied solutions. Two $x$ derivatives multiply $u_n$ by $-n^2$; two $y$ derivatives multiply it by $n^2$. Their sum is zero, and $\sin(nx)$ vanishes on both sides. At the bottom, $\boxed{f_n=0,\quad g_n=e^{-n}\sin(nx)}$. These are smooth fields with compatible side and bottom values.

Step 2: Measure the input and output. For each integer $n$, $\sup_{[0,\pi]}|\sin(nx)|=1$. Hence the bottom norm is $e^{-n}$ and the top norm is $$M_n=\frac{e^{-n}}{n}\sinh n=\frac{1-e^{-2n}}{2n}.$$ Both tend to zero. Thus this sequence by itself does not contradict continuity; the relative amplification is the information needed for the next step.

Step 3: Expose unbounded amplification. The ratio is $M_n/e^{-n}=\sinh n/n\to\infty$. If uniqueness holds in a linear solution class containing these fields, solving the homogeneous linear PDE with data is a linear map. A continuous linear map between the stated normed spaces must obey a uniform bound $\|u(\cdot,1)\|_\infty\le C(\|f\|_\infty+ \|g\|_\infty)$ on its domain. The unbounded ratios rule out that bound. No existence or uniqueness theorem for arbitrary Cauchy data has been assumed.

Step 4: Give a direct discontinuity sequence. Set $w_n=u_n/M_n=\sin(nx)\sinh(ny)/\sinh n$. Its top norm is one, whereas its bottom norm is $n/\sinh n\to 0$. The zero data have the zero solution; uniqueness would force the solution map to select these $w_n$, contradicting continuity at zero. The vertical envelopes are $\sinh(ny)/\sinh n$, not solution traces at one fixed $x$ for every $n$.