Suppose $u\in C^2(\bar B),B$ is the unit ball in $\mathbb R^2,$ satisfies $$\Delta u=f \:\text{ in }B$$ $$\alpha u+\frac{\partial u}{\partial n}=g \:\text{ on } \partial B \ , \alpha>0$$ Where $n$ is the unit outward normal to $B.$If a solution exists then
(1) it is unique.
(2) there are exactly two solutions.
(3) there are exactly three solutions.
(4) there are infinitely many solutions.
Solution:
From the above question, all I understood is $u$ is a continuous function on a unit disk($B$) in $\mathbb R^2$.
- What are the references for the above type of problems?
- How to initiate the above problem.
- Need hint.
First, notice that if the PDE has at least two distinct solutions $u_1$ and $u_2$, then $\theta u_1 + (1-\theta)u_2$ is also a solution for any real number $\theta$. Thus, the only possibility is that the PDE either has one or infinitely many solutions (if is has at least one).
I'll give you a hint for determining if there is only one solution or infinitely many solutions:
Hint: Suppose there are two solutions $u_1$ and $u_2$ to the problem. What PDE does $w = u_1 - u_2$ solve? Can you compute $$\int_B w \Delta w\;dx?$$ What does this tell you?