How many hours is the area sunny each day?

It various within the area. Back close to the trees probably 2-3 hours in the afternoon. Further from the tree 6-7 hours. The poplar trees are deciduous, so in the winter there is full sun.
Cheryl

I think the poplar roots will make it hard for plants to thrive. But have you heard about planting in fabric bags called spin-out bags? These can help keep the roots out. I haven't tried it but I'm planning to. Another option is to plant tough, xeric plants that can deal with having minimal water.

I don't know much about your zone so I can't recommend specific plants, but I'm sure that others can.

Never heard about "spin-out bags"; will do some research. I'm in central California, dry, hot summers (usually in the 90s); no snow in winter (temps don't go much below 30 very often). I water the lawn every day for 20 min in the summer. When I was digging the outline for the planter I could see the water did not soak in more than 2", so I'm sure you are right that the poplar trees are taking a lot of water.
Cheryl

If coneflowers are native to your area, I'd recommend them, mixed in with some native milkweed or other nectar plants to support the pollinators. Native plants are going to have the best chance of thriving in challenging moisture situations. Also, I'd recommend mulch-mowing any leaves that are dropped in fall, and putting the leaves back onto the perennial bed to help improve the moisture retention of the soil, decrease evaporation, and replace nutrients lost during the early growth period of the trees.

Martha

I am not familar with popular roots but I am familiar with growing plants between oak tree roots.

It looks like you are not having trouble growing grass in the area so I do not see a problem growing other plants in the area. You may need to look at some of the other forums to get an idea of what you might like. I would consider some hostas, ferns, geranium, Azelea, Hydrangea.

I gardened at one time in a semi shady property and thought I hated Hostas until I discovered the blues, yellows, and different shapes of the new hostas developed.

The new ferns that are purple, grey, orange are stunning.

I see you are in Zone 9 how about some Abutilon They have some really pretty variegated leaves and long bloom periods.

Because of the tall trees I would probably be more inclined to do bushes so the garden did not look dwarf in comparison to the tall trees.

I would look for bushes with interesting leaf shape, color, and some blooming. So you have a year round color in the area.

Azalea Silver Sword
Abutilon

Here is a link that might be useful: See pics

If the water is in the top 2 in. of soil, then that is where the poplar roots are. I don't know what your plans are for grass removal, but it might not be too hard to get rid of the poplar roots as well. Then if the garden is edged, the poplar roots would cease to be a problem, and you could plant what ever you wanted, and still have less mowing and something nice to look at. I still would go with more xeric plants, like Agastache cana or rupestris or one of the various hybrids that there are, salvias, penstemons at least for the sunny spots. Does your soil drain well? Or some ornamental grasses. You have a lot of good choices there. Harder to think of are things for dry, hot shade. I think back to when I lived in CA and there was a lot of PIttosporum bushes, lantana, StellaD'Oro and Agapanthus. But I was in San Diego.

Some of the large roots are above ground. When I try to mow, the mower blade hits the roots. As far as the grass, right now I am spraying it with Roundup to kill it. This fall, when the weather cools, I will plant the area, then probably cover the dead grass with mulch.
I like the idea of xeroscaping, I also like the ornamental grasses so will incorporate them, as well as some other colorful plants. I don't know about hydrangea, I like it but it may not do too well there. If I can find a small one, I may experiment. I love lantana, and have had good success with agastache (in my former home which had different growing conditions).
I had no idea ferns come in colors, will have to check them out. Lots of good ideas here, thank you all.
Cheryl