LogDCFL-complete problems LogCFL is the set of all languages that are logspace reducible to a context-free language. Similarly, LogDCFL is the set of all languages that are logspace reducible to a deterministic context-free language. See this wikipedia article for some natural LogCFL-complete problems. There are several other interesting LogCFL-complete problems. I could not find any natural LogDCFL-complete problems. Name any natural LogDCFL-complete problem.
The following language is a slight tweak of the usual LogCFL complete one so that it is LogDCFL complete.  The proof can be found in Sudborough's On the Tape Complexity of Deterministic Context-Free Languages.

Let $\Sigma = \{(_1,(_2,)_1, )_2\}$ and $T = \{[,],|\}$.  The following language over $\Sigma \cup T$ is LogDCFL-complete.  $L$ consists of words $$w_0\left[(_1l_1|(_2r_1\right]\ldots\left[(_1l_n|(_2r_n\right]$$ where $w_0, l_i, r_i \in \Sigma^*$ such that there exists $w_1, \ldots, w_n$ with $w_i = (_1l_i$ or $w_i = (_2r_i$ for all $i \geq 1$ and $w_0w_1\ldots w_n$ is parenthetically correct.