Let $X$ be a compact Riemnnian manifold and $\Delta$ the Laplacian. Suppose that $G(x,y)$ be the Green's function of the elliptic operator $\Delta-c$ for a positive constant $c$. I think the following holds by twice integration by parts: $$ \int_{X}G(x,y)(\Delta-c)f(y) dy=\int_{X}(\Delta-c)G(x,y)f(y) dy=\int_X\delta(x,y)f(y)dy=f(x) $$ But a paper I am reading says $\int_{X}G(x,y)(\Delta-c)f(y) dy=-f(x)$. Could anyone point out a mistake in my computation? Or Is the formula a typo?
2026-04-01 04:22:14.1775017334
For Green's function of $\Delta-c$, how to show $\int_{X}G(x,y)(\Delta-c)f(y) dy=\pm f(x)$?
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Your mistake is that $$-\Delta G(x,y) = \delta(x-y)$$