(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let u be harmonic on the bounded region A and continuous on cl(A). Then show that u takes its minimum only on bd(A) unless u is constant.

2. Relevant equations

incase you are used to diffrent notation, cl(a) is clousure bd(A) is boundary

3. The attempt at a solution

By the global maximum principle we have that (let m denote the minimum of u on bd(A))

(i)u(x,y) >= m for (x,y) in A

(ii) if u(x,y)=m for some (x,y) in A, then u is constant.

Assume u is not constant.

then we have that u(x,y)>m.

I dont know where to go from here. I think i still need to show that the only possible place it can take its min is on bd(A). but i have no clue how to do that.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Maximum Modulus Theorem and Harmonic Functions

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**