= 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.
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