How to find Frechet derivate for this functional

Question

My functional is $F(u)=\frac{1}{2}\int (|D_x u|^2 + |D_y u|^2)dxdy + \int fu dxdy$ where $f\in C(\Omega)$, $u\in C_{0}^2(\overline{\Omega})$ and $\Omega \subset R^2$. Well Im reading Elliptic problems in Nonsmooth domains (P.Grisvard) page 2, my question is i need to compute $F(u+\delta)-F(u)$ and try to find some residuous linear in $\delta$? And $\delta$ need to depend of $x,y$? Thank you

Answer 1

Let $u_0, h \in \mathcal{C}^2(\bar{\Omega})$. Then \begin{align} F(u+h) &= \frac{1}{2}\int (|D_x (u+h|)^2 + |D_y (u+h)|^2)dxdy + \int f(u+h) dxdy \\ &= \frac{1}{2}\int (|D_x u|^2+\langle D_xu,D_xh \rangle +|D_xh|^2 + |D_y u|^2+\langle D_yu,D_yh \rangle +|D_yh|^2)dxdy\\&~~~~~+ \int (fu + fh) dxdy \\ &= F(u) + \frac12\int\left(\langle D_xu,D_xh \rangle + \langle D_yu,D_yh \rangle +fh \right)dxdy + \frac12\int |D_xh|^2+|D_yh|^2dxdy \\ &= F(u) + l(h) + o(||h||^2) \end{align} where $l(h) = \frac12\int\left(\langle D_xu,D_xh \rangle + \langle D_yu,D_yh \rangle +fh \right)dxdy$ is linear in $u$. It follows that $$D_uF(h) = \frac12\int\left(\langle D_xu,D_xh \rangle + \langle D_yu,D_yh \rangle +fh \right)dxdy$$