Let $(M,g)$ be a Riemannian manifold, $\Delta$ be the (positive semi-definite) Laplacian on it, and $\lambda_1>0$ its smallest non-zero eigenvalue. There exists $C>0$ such that
$$\|u\|_{L^p} \leq C \|\Delta u \| _{L^p}$$
for all $u \in C^\infty(M)$, $u \perp Ker(\Delta)$. (The condition $u \perp Ker(\Delta)$ is equivalent to having zero mean.) The existence of such a constant $C$ follows from general elliptic theory, though the proofs that I know prove the stronger result $\|u\|_{L^p_2} \leq \tilde{C} \|\Delta u \| _{L^p}$ by contradiction. (Here, $L^p_2$ denotes the $L^p$-Sobolev space with $2$ weak derivatives.)
Question: what is the constant $C$ in the inequality explicitly?
If $p=2$, then $C=\lambda_1^{-1}$. This follows from writing $u$ in a basis of eigenfunctions, which are orthogonal with respect to the $L^2$ inner product. This is reminiscent of using the Rayleigh quotient for computation of the smallest eigenvalue. I don't know what to do about other $p$.
PS: I'm also interested in the answer for $C^{0,\alpha}$ instead of $L^p$.