Hey there,

I need to prove the next statement:

Let $\displaystyle L = A\frac{\partial ^2 }{\partial{x^2}} + 2B \frac{\partial ^2 }{\partial{x} \partial{y}}+C\frac{\partial ^2 }{\partial{y^2}} + D \frac{\partial }{\partial{x}}+E \frac{\partial}{\partial{y}} +F $ be an operator that its coefficients A,B,C,D,E are continiously differentiable twice in the plane.

Show (directly from the definition of an adjoint of an operator) that $\displaystyle L^{**}=L$ .

The definition of an adjoint of an operator is+my problem:

Code:

$\displaystyle L*v = \frac{\partial ^2 }{\partial{x^2}}(Av) + \frac{\partial ^2 }{\partial{x} \partial{y}}(2Bv)+\frac{\partial ^2 }{\partial{y^2}}(Cv) - \frac{\partial }{\partial{x}}(Dv)- \frac{\partial}{\partial{y}}(Ev) +Fv $.
My problem is , that when I substitute L* in the expression for the adjoint, I get partial derivatives of fourth order!!! how can I prove this equality using only this definition?

Hope you'll be able to help me !

Thanks in advance