I recently purchased a house and it has a very nice, solid backyard patio slab. The slab has no cracks whatsoever and is attached well to the house's foundation/slab. However, due to the property being downhill towards the house, the corner of the slab (not the one attached to the house) has sunken a few inches. I would say 2-3, maybe 4 inches at the most. It currently looks ok. It doesn't look or feel sunken, but I have noticed that when I get heavy rain that corner floods a bit. Also, the soil has pushed against it over time that it has formed a bump higher than the floor level. I just built a french drain to make sure the water doesn't come towards the slab or the foundation and now I want to install tile on that patio and I was wondering if I can pour mortar or concrete on that corner to raise it and bring it to level. I will use a floor leveler once it's all leveled, but I feel like I need to fix that corner first. Any recommendations or suggestions on how to raise that corner without having to jack it up, etc? The total area to be raised is about 5 square feet. Thanks in advance!
Since it's draining away from the house, I'd just correct the grade of the soil beyond it so that water isn't trapped and is free to keep flowing away; and leave it be. Dirt is easy to reshape - giant slabs of concrete, less so. If it's grass, peel back the sod, fix the level, roll the sod back down and water it well for a month.
There are ways it can be raised - "mud jacking" is the term of art, basically pumping grout under the slab to raise it hydrualically. Can be expensive, may be difficult to find a local who does it, may cost a lot to bring in a non-local to do it. So I'd pass. If the look of the pergola support is the hill you want to die on, consider that nobody but you will know if you make the pergola supports all the same height and let the top of the pergola not be level ;-). I bet nobody but you notices the difference in the posts if you do make it level...
I recently just watched a YouTube video on doing that to a sidewalk. They actually used many cans of spray foam, and showed later how it was holding up. Possibly search for "Spray can foam level concrete" or "sidewalk" instead of "concrete". They drilled holes just large enough for the tubes from the cans and kept spraying until it raised up. But I also recall he put too much in, because it continued to slowly raise up after some time. https://www.youtube.com/results?search_query=spray+foam+level+sidewalk This shows up some good results for me.