1
$\begingroup$

As part of a problem I would like to use that if u is harmonic, and $u^2((\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y})^2)=0$, then u is constant, we look at the entire plane.

Is this true or not? The problem in solving this is that u may be zero. If u was non-zero we could just say that the partial derivatives was zero, and since the plane is connected this gives that the function is constant. But how do I work around the fact that u may be zero at some points?

Can I use that since every harmonic function is the real part of a holomorphic function, I can in some way look at the function : $f=u+iv$, which is holomorphic?(v is an harmonic conjugate)

$\endgroup$
1
$\begingroup$

If the function is zero on an open set then it is identically zero by the identity theorem (and therefore constant). Thus, if $u\not\equiv 0$ is real-valued, we get $u_x=u_y=0$ except possibly on a set of empty interior.

But since $u_x,u_y$ are also harmonic we get $u_x\equiv u_y\equiv 0$ by another application of the identity theorem. Thus $u$ is constant.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.