 Solution

For compatible $C^2$ boundary functions $a$ and $b$, use exactly the same Volterra series. The factorial estimate holds in the $C^2$ norm after differentiating the integral formula, so the series converges to a $C^2$ function on the full square. It satisfiesand the two boundary values. This directly proves existence. One can equivalently approximate $a,b$ in $C^2$ by compatible analytic functions; the same estimates make their analytic solutions Cauchy in $C^2$.