How to Show u(x,y) and v(x,y) are Constant Throughout D?

[tylerc1991]

Homework Statement

Suppose v is a harmonic conjugate of u in a domain D, and that u is a harmonic conjugate of v in D. Show how it follows that u(x,y) and v(x,y) are constant throughout D.

The Attempt at a Solution

since u is a harmonic conjugate of v, u_xx + u_yy = 0
also, since v is a harmonic conjugate of u, v_xx + v_yy = 0

u_x = v_y => u_xx = v_yx
u_y = -v_x => u_yy = -v_xy

I think that I am going in circles here. Can someone lend a helping hand with this problem? Thank you so much!

 

[Dick]

Hint: in one direction u_x=v_y and u_y=(-v_x). In the other direction v_x=u_y and v_y=(-u_x). Put them together.

 

[tylerc1991]

what do you mean in one direction?

do you mean that since u is a harmonic conjugate then that is one direction, and since v is a harmonic conjugate then that is the other direction?

 

[Dick]

what do you mean in one direction?

do you mean that since u is a harmonic conjugate then that is one direction, and since v is a harmonic conjugate then that is the other direction?

Yes, that's exactly what I mean.