11. Given the function $f(x)=\ln x+\frac{a}{x}(a \in \mathbf{R})$.
(1) Determine the monotonicity of $f(x)$ within its domain;
(2) If $f(x)<2 x^{2}$ holds for all $x$ in $\left(\frac{1}{2},+\infty\right)$, find the range of values for $a$.
(1) $f^{\prime}(x)=\frac{1}{x}-\frac{a}{x^{2}}=\frac{x-a}{x^{2}}$, since $x>0$,

so when $a \leqslant 0$, $f(x)$ is monotonically increasing on $(0,+\infty)$;
when $a>0$, $f(x)$ is monotonically decreasing on $(0, a)$, and monotonically increasing on $(a,+\infty)$.
(2) $\ln x+\frac{a}{x}0$,
thus $g^{\prime}(x)$ is monotonically increasing $\Rightarrow g^{\prime}(x)>g^{\prime}\left(\frac{1}{2}\right)=\frac{1}{2}+\ln 2>0 \Rightarrow g(x)$ is monotonically increasing $\Rightarrow g(x)>g\left(\frac{1}{2}\right)=\frac{1}{4}+\frac{1}{2} \ln 2$. In summary, the range of values for $a$ is $\left(-\infty, \frac{1}{4}+\frac{1}{2} \ln 2\right]$.