Partial Differential Equations Question.

3. Let 2 be a bounded domain in R" with boundary Of and n > 1. Let a(a) > 0 on S and b(x) be continuous on 2. Consider L[u] := Vla(x) Vu] + q(x)u,x En, Blu] = a(2)a(x) Vnu + B(x)u= 0, x E an, where a(x) and 3(x) are continuous and a(x)2 + 3(x)2 > 0 on an. (a) Prove that the operator L is self-adjoint. (b) If an operator L1 associated a boundary condition is defined by n Liu] := a(2) Au+ b(a) > Uxi +q(x ) u, TEL, i=1 Blu] = a(x) a(x) Vnu + B(x)u = 0, x E an, Find conditions on a(x), b(x) and a(x), B(x) such that L1 is self-adjoint