Green's identities

Remarks

The first identity is the divergence theorem applied to the vector field $u \nabla v$, whose divergence is $u \Delta v + \nabla u \cdot \nabla v$. The second identity follows by writing the first one also with $u$ and $v$ exchanged and subtracting. The third identity applies the second one with $v(y) = \Phi(y - x)$ on $\Omega$ minus a small ball $B(x, \varepsilon)$, and lets $\varepsilon \to 0$; the small sphere contributes $u(x)$ in the limit. For the first identity it is enough that $u$ is of class $C^1$. Example: with $u = 1$, the first identity gives $\int_\Omega \Delta v \, dx = \int_{\partial\Omega} \partial v / \partial \nu \, dS$; so a harmonic function has zero total flux through $\partial\Omega$. Example: with $u = v$ harmonic, it gives $\int_\Omega \abs{\nabla u}^2 dx = \int_{\partial\Omega} u \, \partial u / \partial \nu \, dS$, which proves uniqueness for the Dirichlet problem.

Going deeper

For a student

These identities are the basic tool for boundary value problems of Laplace and Poisson type. The first one gives energy estimates and uniqueness of solutions, and it underlies the weak formulation of elliptic equations. The second one shows that the Laplacian is symmetric for functions that vanish on the boundary, which leads to the symmetry of Green's functions. The third one is the representation formula: a function is determined inside $\Omega$ by its Laplacian, its boundary values and its normal derivatives. It leads to the Poisson integral formula once a Green's function is known.