Expansion of the Green's function for divergence form operators
Saïma Khenissy, Yomna Rébaı̈, Dong Ye · Comptes Rendus Mathématique · 2010
We consider the fundamental solution G a of the operator − Δ a = − 1 a ( x ) div ( a ( x ) ∇ ⋅ ) on a bounded smooth domain Ω ⊂ R n ( n ⩾ 2 ), associated to the Dirichlet boundary condition, where a is a positive smooth function on Ω ¯ . In this short Note, we give a precise description of the function G a ( x , y ) . In particular, we define in a unique way its continuous part H a ( x , y ) and we prove that the corresponding Robin's function R a ( x ) = H a ( x , x ) belongs to C ∞ ( Ω ) , although H a ∉ C 1 ( Ω × Ω ) in general.