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My question is related to the Riamann Riemann Mapping Theorem. A function $f$ is biharmonic if $\Delta^2f=0$.

Let $D$ be a simply connected domain in $\mathbb{R}^2$ and denote by $B$ the unit ball. Assume that $f:D\to \mathbb{R}$ is a biharmonic function defined on $D$. Can I find a function $\Phi:B\to D$ such that the composition $f\circ \Phi:B\to \mathbb{R}$ is biharmonic?

If so, what can be said about $\Phi$? Can I choose $\Phi$ to be harmonic?

Note that the composition of a biharmonic function with a harmonic function need not be biharmonic in general!

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# biharmonic morphisms

My question is related to the Riamann Mapping Theorem. A function $f$ is biharmonic if $\Delta^2f=0$.

Let $D$ be a simply connected domain in $\mathbb{R}^2$ and denote by $B$ the unit ball. Assume that $f:D\to \mathbb{R}$ is a biharmonic function defined on $D$. Can I find a function $\Phi:B\to D$ such that the composition $f\circ \Phi:B\to \mathbb{R}$ is biharmonic?

If so, can I choose $\Phi$ to be harmonic?

Note that the composition of a biharmonic function with a harmonic function need not be biharmonic in general!