# functional analysis – Under what conditions is the operator \$ – sum partial_ {x_i} a_ {ij} partial_ {x_j} \$ self-adjoint?

To define:
$$mathcal {L} = – sum_ {i, j} a_ {ij} partial_ {x_i} partial_ {x_j},$$
or $$a_ {i, j} = a_ {j, i}$$. Yes $$a_ {i, j} = delta_ {i, j}$$ the operator is the Laplacian who is known to be self-adjoint in the $$L ^ 2$$ standard.
For other choice of coefficients $$a_ {i, j}$$ – is the operator $$mathcal {L}$$ self assistant in the $$L ^ 2$$ standard?

If we limit our attention to the $$mathbb {R} ^ 3$$ case, we have this
$$int_ Omega v mathcal {L} u , d vec {x} = int _ { partial Omega} v (A nabla u) cdot hat {n} dS – int_ Omega ( A nabla u) cdot nabla vd vec {x},$$
$$int_ Omega mathcal {L} v , d vec {x} = int _ { partial Omega} u (A nabla v) cdot hat {n} dS – int_ Omega (A nabla v) cdot nabla ud vec {x},$$
or $$v in C_c ^ { infty} ( Omega)$$, $$A = (a_ {i, j})$$. Symmetry of $$A$$, the second term in the two equalities is identical, but is there a reason for the first term to be equal in both equalities?