# transferring a PDE estimate from open subset of $\mathbf{R}^n$ to manifold

Suppose $\Omega$ is open bounded domain of $\mathbf{R}^n$ and $L$ is second order linear elliptic partial differential operator on $\Omega$. Fix $f \in L^{2}(\Omega)$, and consider the partial differential equation $$Lu \ge f$$ for say, $u \in H^{1}(\Omega)$. Suppose one proves estimate of the form $$\sup_{\Omega} \ u \le \sup_{\partial \Omega} \ u + C \left\Vert f\right\Vert _{L^{2}(\Omega)}$$ for some constant $C$ depending only on the diameter of $\Omega$.

Now suppose we work on compact manifold $V$ with boundary $\partial V$ instead of open subset of $\mathbf{R}^n$, where we are given a partial differential operator $\mathcal{L}$ on $V$ such that in chart $\Omega$ on $V$ we can write $\mathcal{L} = L$ of form above. Then from above local result can prove a similar result: for $f \in L^2(V)$ and $u\in H^{1}(V)$ if $$\mathcal{L}u \ge f$$ then $$\sup_{V} \ u \le \sup_{\partial V} \ u + C \left\Vert f\right\Vert _{L^{2}(V)}$$.

Question is, what will the constant $C$ now depend on?

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