I just want the brain fog to go away, and the depression to lift. But I think I’ll always be depressed, and even more so the more I do this. So, color me depressed. I try that bullshit positive affirmations, and they work to an extent, but some days just suck. Like today. When I couldn’t prove immediately that a uniformly convergent sequence of harmonic functions is harmonic. I thought of covering the space (topology in action), and of course, I can’t guarantee that the radii of each harmonic function where the mean value property holds even *forms* a cover of the space. It took a simple picture of disjoint balls in a domain to see this, and immediately demolished the proof I wrote. Why I thought I could take a cover was some… mental tethering to methods I know in a different discipline.

Either way, the proof now is to consider all of the harmonic functions that are defined on these r_i contained entirely in a domain, not necessarily overlapping and definitely not covering the space. Construct an extension function \tilde{u} that will be a Poisson kernel, which is harmonic. Since the function is harmonic, take the boundary of the domain its defined and the conditions on the boundary determine the function on the interior.

But here is where I get confused. Supposedly maximum principle applied to this will give us that this function is unique and is harmonic, but I’m not sure how these sequence of harmonic functions in the interior will work out.

So we have these harmonic functions defined on the interior, and the boundary conditions of an extended function… and something about solutions being unique.

Alright, I was supposed to be lamenting about how I’m bad at math, not actually proving that harmonic functions behave well under uniform convergence. I’ll save the real proof of this for the complex analysis section.

The sun is out today. Maybe I’ll be able to finish my measure theory problem set. Alone, unfortunately. Unless Joon is around. Everyone else is useless as an analyst.

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