= Solution
The zero values at $x=0,a$ select $X_n=\sin(n\pi x/a)$, and the zero value at $y=0$ selects $Y_n=\sinh(n\pi y/a)$. Expanding the constant upper-edge value as a <Fourier sine series> gives coefficients $b_n=2F[1-(-1)^n]/(n\pi)$. The resulting <harmonic function in a rectangle with one constant boundary value> is
$$
\boxed{f(x,y)=\frac{4F}{\pi}\sum_{m=0}^\infty\frac{\sin((2m+1)\pi x/a)}{2m+1}\frac{\sinh((2m+1)\pi y/a)}{\sinh((2m+1)\pi)},\qquad0<x,y<a.}
$$
Each term satisfies <Laplace's equation> and the three zero boundary values. At $y=a$, the series equals $F$ for $0<x<a$. On compact subsets of the interior it and its derivatives converge exponentially, which justifies the termwise equation.
For $F\ne0$, the top-corner boundary prescriptions conflict: a corner cannot be both zero on a vertical edge and $F$ on the upper edge. The bounded interior solution has the specified limits along the open sides; it has no single continuous extension at those two corners. Values assigned only at the isolated corners do not change the interior solution.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1-square-harmonic-extension.png]
{title=Harmonic extension from one heated edge of a square}
{description=Original normalized solution $f/F$, with the upper open edge fixed at one and the other three open edges at zero. The top corner circles mark incompatible side limits, not prescribed continuous corner temperatures.}
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