Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$.

Suppose one knows that there is some $C_1,C_2 \in R$ such that $ x \mapsto C_1 | \nabla E(x)|^2 + C_2 \Delta E(x)$ is constant on $ \partial \Omega$. I am interested in what one can conclude about $\Omega$ and (or) $E$.

Suppose $ \Omega$ is a ball centered at the origin and $E$ is radial. Then is satisfies the above. (of course one can take a ball and extend in one more direction and let $E$ be independent of this extra dimension and still arrive at the same thing). Is this the only option?