 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 isEach 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 $F\ne 0$, 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.