Partial Differential Equations Question.

3. Let 2 be a bounded domain in R" with boundary On and n > 1. Let a(x) > 0 on n and b(x) be continuous on 2. Consider Lu] : = Vla(x) Vu] + q(2)u, x En, Blu] = a(2)a(x) Vnu + B(x)u = 0, x Ean, where a(x) and S(x) are continuous and a(x)2 + 3(x)2 > 0 on 20. (a) Prove that the operator L is self-adjoint. (b) If an operator L1 associated a boundary condition is defined by n LI[u] := a(x) Au + b(x) Luxi t q(x)u, x En, i=1 Blu] = a(x)a(x) Vnu+ B(x)u = 0, x Ean, Find conditions on a(x), b(x) and a (x), B(x) such that L1 is self-adjoint. 4. Let L be the same operator as in the problem 3(a). Find the sufficient condition for q(x), a(x), B(x) such that all eigenvalues for the problem L[u] = -do(x)u subject to the same boundary condition in the problem (3) are positive, where o(x) > 0 on 2