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I would like to classify functions $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $$ \nabla \cdot \left( \phi \frac{\nabla\phi}{|\nabla\phi|} \right) = \text{const}. $$

The only examples I know are $\phi(x,y)=(x^2+y^2)^{1/2}$ and $\phi(x,y) = cx$, $\phi(x,y) = dy$. Are there more examples of such $\phi$ "in between''?

Injee
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    Any linear combination $\phi(x,y) = ax + by$ is also a solution. $\phi(x,y) = \frac{1}{\sqrt{x^2+y^2}}$ also works if the origin is excluded from the domain. – Winther Jun 22 '16 at 20:28
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    Amazingly, any linear combination of $\sqrt{x^2+y^2}$ and $1\over\sqrt{x^2+y^2}$ also seems to work. – Ivan Neretin Jun 22 '16 at 21:31
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    @IvanNeretin For $\frac{a}{\sqrt{t^2+x^2}}+b \sqrt{t^2+x^2}$ the operator in the lhs gives $-2 b \operatorname{sgn}\left(a-b \left(t^2+x^2\right)\right)$, so actually it's a solution for $ab\le0$. – Andrew Jun 29 '16 at 20:04
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    Don't know if that helps, but by Poincare's lemma $\nabla \cdot u = 0$ on $\mathbb{R}^2$ implies $u = \nabla^\bot \psi$, where $\nabla^\bot = (-\partial_y,\partial_x)$. So any solution has to be of the form $\phi \nabla \phi / |\nabla\phi| = \nabla^\bot \psi + c f$, where $f$ is a function with $\nabla f = const$ which can be fixed as something nice beforehand. – mlk Jul 03 '16 at 12:43

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