Lecture notes on Green function on a Remannian Manifold

L = ∆g + aI Here a ∈ L∞(M) and ∆gu = −divg(∇u) (Or for simplicity you can choose a ∈ C∞). And we define the kernel Ka (depending on the constant a) of the differential operator L: Ka = KerL ∩W 1,2 0 (M) It is obvious that if the differential operator is coersive, then the Kernel Ka = 0. Definition 1. (Green Function) Define G : M̄×M̄\diag{M̄} → R is a Green function for L with Dirichelet Boundary condition if ∀x ∈M : (i) G(x, ·) = Gx ∈ L(M) (ii) Gx ⊥ Ka (iii) if φ ∈ C c (M̄), then: ∫