Consider the 2d Laplace equation $$\Delta u=0$$ on the upper half plane $$\{(x,y)| y \geq 0\}$$
I know the problem is well posed if you specify Dirichlet boundary conditions at $y=0$, and a suitable boundedness as $y\to\infty$: $$u(x,0)=f(x)$$
I know it is ill-posed if you specify both the function and its derivative at $y=0$: $$u(x,0)=f(x) \,, \quad u_y(x,0)=g(x) $$
I need to impose more unconventional boundary conditions: $$u(0,y)=h_1(y) \,, \qquad u_x(0,y)=h_2(y)\,, \quad y\geq 0$$ namely specifying both the function and its derivative at $x=0$.
Is this problem well-posed?
Any help or references to the appropriate literature would be insanely helpful, thanks in advance!
The first problem with the Dirichlet condition is not well-posed. The function $v \colon (x,y) \mapsto y$ verifies $\Delta v = 0$ on $\{(x,y) \in \mathbb R^2 \colon y > 0\}$ and $v(\cdot,y=0) = 0$.
So for every solution $u$ of your problem, $u+v$ is also a solution.
Conditions at $y \to + \infty$ are lacking.