13 - Maximum Principle

Properties of Dirichlet Problem corollary

For Dirichlet Problem for $u$ with boundary of $D$

$$\begin{array}{rl}{u}_{xx}+u_{yy}& =f(x,y)\text{ in }D\\ u& =g(x,y)\text{ on }\partial D\end{array}$$

then

1. If the solution exists it is unique

2. Solution has continuous dependence on data

Proof

1. Suppose that $u_1,u_2$ are two solutions so $u=u_1-u_2$ satisfies

$$\begin{array}{rl}{u}_{xx}+u_{yy}& =0\text{ in }D\\ u& =0\text{ on }\partial D\end{array}$$

So by The Maximum Principle for the Laplacian then
the maximum and minimum of $u$ occur on $\partial D$ hence $u\le 0$ and $u\ge 0$ on $D$ so

$$u=0\text{ in }D$$

2. Need to prove that for all $ϵ>0$ there exists $\delta >0$ such that if

$$u_i,i=1,2\text{ are solutions }\text{ with boundary data }{g}_i$$

then

$$\underset{(x,y)\in \partial D}{\sup }|g_1(x,y)-g_2(x,y)|<\delta ⟹\underset{(x,y)\in D}{\sup }|u_1(x,y)-u_2(x,y)|<ϵ$$

By linearity $u=u_1-u_2$ satisfie

$$\begin{array}{rlrl}{u}_{xx}+u_{yy}& =0& & \text{ in }D\\ u& =g_1-g_2& & \text{ on }\partial D\end{array}$$

Using maximum principle we get that

$$u\le \underset{(x,y)\in \partial D}{\max }(g_1-g_2)$$

So applying same result to $-u$ we get

$$-u\le \underset{(x,y)\in \partial D}{\max }-(g_1-g_2)$$

Hence in $D$

$$|u_1-u_2|\le \underset{(x,y)\in \partial D}{\max }|g_1-g_2|$$

so take $\delta =ϵ$

Property of Heat Problem

Consider IBVP

where $u(x,t)$ satisfies

$$u_t-u_{xx}\le 0$$

in a region $D_{\tau }$ bounded by

1. lines $t=0$ and $t=\tau >0$
2. two non-intersecting smooth curves $C_1$ and $C_2$ which are nowhere parallel to $x-$axis

with $u$ given $\partial D_{\tau }\{t=\tau \}$ then

$$\text{if solution exists it is unique and depends continuously on initial data}$$