I am looking for a reference where the following problem is discussed:

$u \in C^{\infty}(I^2)$ so that

  • $\Delta u = 0$
  • $u(0,y) = f(y)$, $u(1,y) = g(y)$, $u(x,0) = h(x)$
  • $\nabla u(x,0) \cdot \hat{n} (x,0) = a(x)$

Assume $f'(0) = a(0)$ and $g'(0) = a(1)$.

That is, $u$ is harmonic in $I^2 = [0,1]^2$, and subject to certain mixed and overlapping boundary conditions -- Dirichlet conditions on the three sides, and Neumann conditions on one of those sides, and the fourth side is left unspecified. (In other words, on one side there is are no conditions, and on two opposite sides there is a Dirichlet boundary condition, and on one side there is a Cauchy boundary condition.)

Specifically, I am interested in the conditions on $f,g,a,b$ necessary for a solution to exist, and to be unique, and for a (reasonably) explicit solution.

Some appropriate terms to search would also be very welcome.

P.S. I'm not especially committed to the Laplace equation -- it's the boundary conditions along with existence and uniqueness that are more important for my purpose right now. I'd be interested in replacing $\Delta u = 0$ by another PDE in order to get a problem that has existence and uniqueness for these boundary conditions - though the problem as I formulated it above has certain advantages from my point of view.


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