# Help me prove that functions u and v must be constant in D

1. Jun 13, 2004

### Ed Quanta

Ok, so let us suppose that functions u(x,y) and v(x,y) are harmonic in a domain D. Thus uxx+uyy=0, and vxx+vyy=0. In addition, v is a harmonic conjugate to u. Thus ux=vy, and uy=-vx.

*Note ux= partial derivative of u with respect to x, vx=partial derivative of v with respect to x, and so on.

I have to use all of this information to show that functions u and v must be constant throughout domain D.

2. Jun 14, 2004

### HallsofIvy

Staff Emeritus
You can't prove it: it's not true.
From ux= vy, it follows that uxx= vyx and from uy= -vx it follows that uyy= -vxy so that any "harmonic conjugate" functions MUST be harmonic. In particular, if u= ax+ by and v= ay- bx, then uxx= 0, vyy= 0, ux= vy, and uy= -vx for all x,y but u and v are NOT constant. Are you sure you haven't misplaced a sign in your problem?

3. Jul 24, 2004

### mathwonk

I think what is true is that if a harmonic function has a local maximum at a point in an open domain then it is constant, and this follows from the integral formula for harmonic functions, analogous to cauchy's integral formula for holomorphic functions.