In $2$ dimension, take $-\Delta u=0$ on $\{(x,y\},y\geq 0\}$ with $u(x,y=0)=f(x)$, $u_y(x,y=0)=g(x)$ where $f$ and $g$ are smooth function. I want to justify whether this problem is well posed.
My first question is, what do we mean by a problem is well-posed? In my opinion a problem is well-posed if this problem
$(1)$ has a solution
$(2)$ the solution is unique.
$(3)$ the solution continuously depends on the data, i.e., the boundary value in my problem.
But my friends tell me that I don't need uniqueness, only existence is enough for well-posed problem. I confused, do I need uniqueness or not?
Now, go back to my problem. The book states that this problem is not well-posed and gives an example such that $$ \frac{1}{n^2} e^{n\epsilon y}\sin(nx) $$ But I failed to see why this example give me the contradiction...
Any help is really welcome!