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? 



comparing Laplacian and gradient of function on boundary
For any smooth $\Omega$ and $E$ one can perturb $E$ so that the desired conditions hold. Indeed, let $d$ be a global smooth function agreeing with the distance from $\partial\Omega$ in a neighborhood of the boundary. Let $F$ be any smooth function on the boundary, globally extended so that near the boundary it is constant on normal lines. Finally, consider $G = F(x)d^2(x)/2$. Then $G = |\nabla G| = 0$ on the boundary. Tangential to the boundary, $G$ separates at most quartically from zero so its tangential Laplace is zero, so on the boundary $\Delta G = F$. Choosing $F$ appropriately one sees that $E + G$ satisfies the desired conditions.


