Keeping a lawn looking nice in the shade is tough. We have a dense tree next to our porch, and the solution we've settled on is overseeding with a "shade" grass seed every year or two. You don't need to rip out the existing grass, just scatter the seed in the thin/bare spots and lightly rake in. Fall is a good time to seed, but do it while your sprinklers are still turned on so it can get some water.

If the ground remains consistently damp, then you may have a problem with drainage in that area. Just overseeding every year is not going to help much if that is the problem and it is not fixed or addressed to correct what is causing the area not to drain properly.

It's the shade. Those trees are so close together! Were there a drainage problem they wouldn't be so healthy looking either. Yearly topseeding in the fall will help and should be done with the open lawn as well.

If you have a favorite spot, you can lay 2x2 concrete pavers flush with the lawn surface. Your chair will be stable, not sinking in, and you can mow right over them. They are also available in 16"x16" and in quite a few colors; so you can more or less blend them in with your grass.