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More or less, this is meant in the sense of "morally" hard. However, many people work on / with the obstacle problem (also in terms of numerical approximation). Therefore, I think there is no explicit formula for arbitrary $u$.
It's not totally clear what you mean by projection. Of course, you can take the element in the positive cone with the smallest distance (which is unique for $p \in (1,\infty)$). But I don't think that this is your question. In the case $k = 1$, $p = 2$, this projection is given by the obstacle problem
