11. Let $a, b, c, d$ be real numbers, find
$$
a^{2}+b^{2}+c^{2}+d^{2}+a b+a c+a d+b c+b d+c d+a+b+c+d
$$

the minimum value.
Solve $\left\{\begin{array}{l}\frac{\partial F}{\partial a}=2 a+b+c+d+1=0 \\ \frac{\partial F}{\partial b}=2 b+a+c+d+1=0 \\ \frac{\partial F}{\partial c}=2 c+a+b+d+1=0 \\ \frac{\partial F}{\partial d}=2 d+a+b+c+1=0\end{array}\right.$, so when $a=b=c=d=-\frac{1}{5}$, $F(a, b, c, d)_{\min }=-\frac{2}{5}$.