Nice. My immediate instinct is to do this with the divergence theorem. The Laplacian is the divergence of the gradient, so as soon as you say “average value on a shell” I’m immediately getting flashbacks to Physics 260 and thinking “well the integral of the Laplacian inside the shell is equal to the integral of the gradient through the shell—is there some way to get the normal component of the gradient out of the difference between the value at the center minus at the surface?”
Nice. My immediate instinct is to do this with the divergence theorem. The Laplacian is the divergence of the gradient, so as soon as you say “average value on a shell” I’m immediately getting flashbacks to Physics 260 and thinking “well the integral of the Laplacian inside the shell is equal to the integral of the gradient through the shell—is there some way to get the normal component of the gradient out of the difference between the value at the center minus at the surface?”