It looks like creeping charlie (Glechoma hederacea) It is almost impossible to get rid of, or so it seems. Chemicals don't seem to faze in a lot, so I pull as much as possible. I almost like it, actually. If it wasn't so naughty and determined to take over EVERYTHING I'd be tempted to leave it. I just pull and pull and pull, then pull some more. It thrives and will send roots from every node that touches the ground.

You have 3 different weeds there....1 is creeping charlie, another is common violet and the 3rd looks to be some sort of a campanula...rapunculis likely....has a root like a carrot and regenerates whole plants from the tiniest scrap of root.
You can pull or use poisions....but neither respond well to poision unless you use a pre emergant as well.
Linda C

I find (and I think this is what Linda was referencing when she suggested pre-emergents) that when herbicides appear to not kill weeds often they do kill the weed that got sprayed, but new ones grow from seeds quickly so the overall number of weeds in the lawn appears to stay the same.

That is where the pre-emergent comes in, it prevents germination or kills the new sprout before you ever see it. It is a must have for the heavy seeders. The alternative is to go out and pull/spray every couple weeks to try and get control.

But....it takes industrial strength to kill violets....not considering the seeding. Spray normal stuff and the leaves just get a little brown around the edges....and they're baaaaack!
Linda C

A possible is to heavily lime the area. I know when I had an overload of sorrel, changing the soil pH in the growing area certainly made it's life very difficult.

If you grow fine azaleas/rhodos then you probably have a soil pH between 6.5 and 5. If you can get it over 7 and hold it there for a few months that could help.

The other areas where you may be able to intervene are: in the amount of moisture available to the plants. Going from reliable wet to rare and droughty can be useful; and the amount of sunlight the area gets. And gross feeding through adding compost can change the environment enough to suppress the plants. [Though some are thug enough to just lick their collective chops and go for more...:-( ]

On the creeping charlie - the only thing I have found that works is Mule Team Borax and water. My yard used to be full of creeping charlie but after spraying the this mixture, I have NO creeping charlie. (The neighbors have plenty so I have to watch the fence lines for runners!!) I used a mix of 5 tablespoons of Mule Team Borax to one gallon of water and put it in a pump sprayer. (actually I mix five gallons in my sprayer) This mix will not harm the grass unless you mix it very strong. Within 24 hours (on a warm, sunny day) the creeping charlie was shriveling up and was gone in no time. Make sure you spray all of it.

Be careful when using borax for this purpose. The linked article outlines the issue well.

Here is a link that might be useful: U of Illinois