The short answer is that the definite article should be present in both cases.

As a title : Knowledge of day : if we add "the", the meaning would be knowledge of today. If we had "a" we are talking about a specific day in time, either past or future. Knowledge of day sounds more like "знание о дне" or "знание о днях" about what there is to know about "days" in general - that's a bit weird though meaning-wise. Do not lean on door - это нормально, the "the" could be added but it is superfluous, not needed, so it is common to see it written this way, if you want to be very precise you can add it, but if everyone knows which door you are talking about, not needed, короче - лучше.