it's hard to control a dog's urination, because it's not only a function of elimination, but also used by dogs to mark their territories. Very often, the dog would ration it's volume, few drops here and there, so it can mark more "land marks", haha :)) This type of behavior is printed in their genes. However, it's absolutely the owner's responsibilities to clean up their dog's "dodo".
There are a few things you can do:
1. Put a clear sign on your lawn, maybe attach a small box containing a roll of plastic bags, like those boxes we've seen at the entrance of public parks.
2. If you have a tradition of hosting a christmas party for the neighbors, maybe you can invite this family in? Try to get to know them better, after all you are neighbors. Or just a small christmas gift, i do that every yr for my neighbors. You never know, good neighbors could make youlife easier.