# comparing Laplacian and gradient of function on boundary

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?

• Probably one can get the boundary Laplacian of $E$ if one knows the mean curvature of $\partial\Omega$. Oct 17, 2015 at 17:38

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.
• @ Connor Mooney. Thank you very much for the answer. Let me adjust the questions slightly (and maybe your answer answers this too... i too dumb to tell). Let $0<E$ is smooth and lets assume $E=0$ on the boundary of $\Omega$. I am curious if one can have a situation like $C_1 | \nabla E(x)|^2 + C_2 \Delta E(x)$ constant on every level set of $E$ (there the $C_i$ can depend on which level set but no more). thanks Oct 18, 2015 at 18:26
• @Math604: For any $\Omega$, the solution to the eigenvalue problem $\Delta E = -\lambda E$, $E|_{\partial \Omega} = 0$ where $\lambda$ is the smallest eigenvalue would satisfy these conditions with $C_1 = 0$. More generally, for your second question, $E$ would have to solve a PDE of the form $\Delta E + f(E)|\nabla E|^2 = g(E)$. Oct 18, 2015 at 18:48