I'm trying to solve this question on self-adjoint in complex numbers. I'm stuck and need help. Consider $ L =\frac{d^3}{dx^{3}}$ to be a linear operator which acts on a function from $[0,1]$ to $\Bbb R$. Given the boundary conditions
x(0) =1,\quad x'(0) - 2x''(1)=0, \quad x(1) -x'(1) = 0, where all the quantities are real. An inner product is defined as $ \langle u(x), v(x) \rangle =\int_{0}^{1} u(x)v(x) dx.$ I'm to determine whether $L$ is self-adjoint.
This is what I have done so far , please help me as to what to do next.
\langle u(x), Lv(x) \rangle = u(x)v''(x)|_{0}^{1} - u'(x)v'(x)|_{0}^{1} + u''(x)v(x)|_{0}^{1} - \langle Lu(x), v(x) \rangle. That is after integrating by parts three times. Now, I don't know what to do since after plugging the upper and lower limits, there is nothing left since . Any guidelines?