In this post, we shall see the maximum principles for subharmonic ( the Laplacian is nonnegative) functions( twice differentiable with continuous second derivative). As an application, we shall see the uniqueness of a solution to the Dirichlet problem. The Dirichlet problem is:

Suppose is an open bounded subset of . Let be a continuous function defined on , the boundary of . Does there exist a function that solves

Historically, the Dirichlet problem has been important to solve PDEs that arise naturally in Physics and engineering. But as mathematicians, let us not worry about that. For us, it is important to know when a solution to the DP exists and when its unique. Curiously, a sufficient condition on the existence of a solution is how looks! For example, if then exists! In the following part, we shall show using the “Maximum Principle” that if an exists, then it must be unique.

**The Maximum Principle**

**Theorem 1 (Strong form)** Suppose is an open subset of and in . Then cannot attain a maximum in .

**Idea of proof:**

By contradiction: Suppose attains its maximum at . Find an such that and let be . Show that and . Add over all to get , a contradiction.

In the weak form, we assume the function to be subharmonic, i.e., .

**Theorem 2 ( Weak form)** Suppose is an open **bounded** subset of . (So now, both and are compact sets). If in , then

**Idea of proof:** We use the trick.

Define

Show that , and use the strong form of the maximum principle.

**Proof of uniqueness in the DP:**

Suppose that and both solve the DP equation (1). Define . Then, in and on . Hence by the weak form of the maximum principle, in . Since we could have taken to be instead, it follows that , by which in and uniqueness has been established.

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February 22, 2012 at 05:47

Nisha iyerHey abhi..did read this post…although full googly gaya..write some nic experience also so that we people who are little naive to this algebraic equation can also enjoy :):)

February 21, 2015 at 19:26

RMplz can u help me to prove the uniqueness of this ODE: on (0,r) $-u”=(1-u^2)u$ u(0)=0 and u(r)=s (s is strictly positive), ive worked hard and searched a lot to find the proof and all that i got are some hints on the proof. i really need the proof details to complete an important part of my thesis,thanks in advance

February 21, 2015 at 20:18

Abhishek ParabPerhaps math.stackexchange would be a good place to ask this question.