12 - Well Posed Problems for PDEs

Well Posed PDE

Problem consisting of a PDE with data is well posed if solution

1. exists
2. is unique
3. depends continuously on data

Solution Depends Continuously on Data

Let $u(x,y)$ be a solution of a PDE in a bounded subset of plane $D$ with $u$ given on curve $\Gamma$

Solution $u(x,y)$ depends continuously on data if

$\forall ϵ>0\exists \delta >0$ such that if $u_i,i=1,2$ are solutions with $u_i=f_i$ on $\Gamma$ then

$$\underset{\Gamma }{\sup }|f_1-f_2|<\delta ⟹\underset{D}{\sup }|u_1-u_2|<ϵ$$

Models from Prelims