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

[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.

 

[HallsofIvy]

You can't prove it: it's not true.
From u_x= v_y, it follows that u_xx= v_yx and from u_y= -v_x it follows that u_yy= -v_xy so that any "harmonic conjugate" functions MUST be harmonic. In particular, if u= ax+ by and v= ay- bx, then u_xx= 0, v_yy= 0, u_x= v_y, and u_y= -v_x for all x,y but u and v are NOT constant. Are you sure you haven't misplaced a sign in your problem?

 

[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.