 Forced zero trace at a quadratically degenerate boundary

For the bounded continuous solutions in the sine-mode expansion for a quadratically degenerate elliptic equation, every surviving mode tends to zero at $y=0$. The estimate $p_n^{+}>n$ bounds the absolute sum by a constant times $y/(1-y)$, uniformly in $x$. The lower boundary trace must therefore be zero. Continuous boundary data prescribing a nonzero lower trace and zero vertical sides are incompatible with the equation.