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I need help for finding the extremal of: $$J[u]=\int\int_D (u_x^2+u_y^2) dxdy$$ $D$ is the unit disc i.e. $x^2+y^2 \leq 1.$ My boundary condition is $$u(\cos\theta, \sin\theta)=\sin(n\theta), \ \ 0\leq \theta \leq 2\pi$$ I have been using polar coordinates so far, please could you advise me how to approach this problem. Thanks! |
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Interpreting $(x,y)$ as the complex plane with $z= x+i y$. The Euler-Lagrange equation $$ \Delta u =0$$ are solved by real and imaginary parts of holomorphic functions. Note that $f(z) = z^n$ is holomorphic in the unit disc and its imaginary part fulfills the boundary condition. So the solution is $$u(x,y)= \text{Im} z^n = r^{n} \sin( n \theta)$$ with $r=\sqrt{x^2+y^2}$ and $\tan\theta= y/x$. |
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