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I came across the following problem which is as follows:

"A bounded harmonic function in the unit disc centered at origin and taking the value sin$2\theta$ on the boundary is $r^{2}$sin$2\theta.$" I have to determine whether this statement is true or false.

I know that a bounded harmonic function is constant on $\mathbb C.$ But could this result be of any use? Can someone point me in the right direction?Thanks in advance for your time.

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No, information about a function on all of $\mathbb C$ won't help with this. – GEdgar Feb 2 at 14:48
this is known as the dirichlet problem (for harmonic function). in your case, you need to 1. check that it is indeed a solution and 2. justift the uniqueness (you can do that with the maximum modulus principle). see also en.wikipedia.org/wiki/Poisson_kernel for a generalization – Glougloubarbaki Feb 2 at 14:53
@Glougloubarbaki If they check that it is a solution, why would they need to check that it is unique? It looks to me that they need to justify that it is harmonic and then they are done. user33540 Just use the polar form of Laplace's equation and it will be easy to check. – Matt Feb 2 at 17:19
@Matt Thanks a lot sir.I see that the function $u(r,\theta)=r^2 sin(2\theta)$ satisfies the Laplace's equation and so I think the above statement is true. – user33640 Feb 2 at 17:40
"A bounded harmonic function in the unit disc centered at origin and taking the value sin2θ on the boundary is r2sin2θ." ok, that is a case where grammar matters : can the sentence not be understood as "if a function satisfies these hypotheses, then it must be this one" ? – Glougloubarbaki Feb 2 at 20:10

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