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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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