Hi! I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev Embedding Theorems (more functional analysis) and one on variational elliptic problems in PDEs (which as I understand will focus more on proving the existence and uniqueness of solutions). I'm already studying Sobolev Spaces and its properties and the direction I take after this will differ based on what I choose. Unfortunately, both interest me equally which is why this continues to be a dificult decision to make.

So, I want to ask what writing or doing research on these topics are like? Is there a topic that's more important to learn as early as now for a student wanting to pursue graduate studies?

Thank you so much!
Can you let me know what resources you'll be using? I am interested to know more about these topics
Well-posedness of PDE's in general was/is and probaly will be a hot topic in the foreseeable future. Hyperbolic PDE's or general Fokker-Planck-ish stuff is quite hard to analyze for well-posedness in general settings and there are definitly some open questions still. Less so in elliptic equations from what I know/hear but I might be wrong, as I am working on hyperbolic/transport stuff.

What exactly will your work on Sobolev inequalities involve? This might be interesting as Bachelor thesis to get involved in the general setting and getting a feeling for the literature out there. However this does not seem like a topic with a lot of active research needed/happening.
Am I missing something about your topics? You can't really discuss existence and uniqueness of elliptic PDEs without sobolev spaces/inequalities. Your two topics go hand in hand to me, you can't properly handle one without the other.

An example I like to use: take the basic heat equation on R^n. Unless you restrict the space of solutions to some sobolev space you lose uniqueness: there are "solutions" beginning with initial condition 0 which are immediately non-zero.
Sobolev embedding is maybe two big workhorse theorems and a lot of more niche embedding-like theorems.

Even the classical theory of well-posedness of second order elliptic pde is a very rich landscape.

My point is that the topics aren't even remotely comparable in scope, and what you choose to do depends more on how long and detailed the thesis needs to be rather than on what is interesting.
Research in these fields might be out of reach for an undergrad. These are mature fields and most of the current work in what you’ve mentioned will either be too specialized or too abstract.

For example, there is a theory for Sobolev spaces on sets that don’t have a Euclidean structure, e.g. metric measure spaces (you cannot “add” points or “scale” them like you can for points in Rn). There are quite a lot of steps to go through from someone starting to learn about Sobolev spaces for subsets of Rn to this.

With regards to well-posedness, researchers would often prove it for a specific model that is not quite the same as the ones you see in classical literature. For example, you might consider models coming from the sciences with very specific terms that, because of strong connections to either  physics, chemistry, or biology, might say something more about the solution than what classical theory will tell you.

There are plenty more to look at for elliptic PDEs aside from existence and uniqueness. Mathematicians are also interested about the regularity/smoothness of their solutions. These depend on the structure of the equations you’re studying, the regularity of data and the coefficients of your equations, and the regularity of the domain boundary (if you’re working on bounded domains or domains with boundaries). There is also an entirely different approach for fully nonlinear elliptic PDEs that do not use Sobolev spaces. We call them viscosity solutions, and there is a jungle of problems and research directions to go towards to.

So going back to your original post, research in the sense of contributing something new and publishing these results, is most likely out of reach for most undergrads.

What you can do as a thesis, and ask your department about this, is do an expository paper on these topics. The theory of Sobolev spaces on Rn alone is a lot for an undergrad to learn and write about.

Hope that helps. Good luck! Feel free to dm me if you have questions :)
Undergraduate theses must have mentors - ask them first. Just exploring as much as you can about Sobolev embedding within a semester and perhaps reproduce/rewrite some of the basic theorems would be a neat undergrad thesis.

On the other hand, getting some exposure in proving existence/uniqueness before grad school might help you when you inevitably go through Evan's and then some more on seminars. Integration by parts is not inherently hard but when one estimate involves breaking an integral into 4 parts and each of their estimate takes several pages of IBP/Interpolation inequalities, you will want to pull your hair out (looking at you Vlasov-Poisson-Maxwell ). So getting used to this feeling early in undergrad should be nice.