Is this Laplace BVP well posed? If not, why?

Special solutions of the Laplace equation that obey all of the OP's boundary conditions are: $$ \phi_k(x,y)=e^{-k\pi/a.y}\cos\left(k\frac{\pi}{a}x\right) \quad \mbox{with} \; k=1,2,3,\cdots $$ The Laplace equation is linear, hence a more general solution is: $$ \phi(x,y) = \sum_{k=1}^n c_k\,\phi_k(x,y) \quad \mbox{with} \\ \phi(x,0) = \frac{1}{2}c_0 + \sum_{k=1}^n c_k \cos\left(k\frac{\pi}{a}x\right) $$ The latter expression can be interpreted as a (partial) Fourier expansion of the even periodic continuation of $\,\phi(x,0)=f(x)\,$: $$ c_k = \frac{2}{a} \int_0^a f(x)\,\cos\left(k\frac{\pi}{a}x\right)\,dx $$ In particular: $$ \frac{1}{2}c_0 = \frac{1}{a} \int_0^a f(x)\,dx = 0 $$

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