I'm a first year phd student in Germany. I've started my phd study one year ago and I'm currently confused about the topic I've chosen. The program is in the area of PDEs, and actually I didn't learn much about PDEs in my master study, only for instance the solution of the four fundamental types of PDEs and some Sobolev space theory. It was a challenge for me to learn the PDE theory at the beginning, since one doesn't only need to know a lot of things involving mathematics, but should also learn something about the physical modelling. This was a little bit hard for me since my minor subject was economics. But after the one year learning I think studying physical models is also very attractive and interesting. But as more as I learned, I know one can generally not conclude a very central theory for all the models, which causes a problem to my phd topic. My topic is regularity theory for a general setting of physical models. As one knows, the general trick is to consider the Nirenberg-difference quotient in such regularity theory, supposing the difference quotient is still in the space of test function. But now, if we give some first order differential conditions to the test functions, for instance $\nabla\cdot\phi=0$ or $\nabla\times\phi=0$ for a test function $\phi$, then one needs not to have that the difference quotient is still divergence free. My task is to find a trick to solve these problems. But as more as I learned, I have more and more confusion:

1. I can not find any models which use restricted test functions. The restrictions here are mostly given by divergence or curl free conditions. But these conditions are usually to solutions but not to test functions. Examples are the Maxwell's equations and piezoelectric model.

2. My advisor asked me to give a generalization which concludes a class of such problems, but as you know, PDE questions are usually formulated very differently. Indeed, I can give some sort of generalization to the conditions, but it seems that these generalization does not make any sense in physics. I can not see that a generalization makes a contribution to physical models, since in general only the gradient, divergence or curl appear in the most models and have a clear physical meaning.

I will have a talk to my advisor in next week, eventually I will ask her to give me some real world models that relate to my topic, which seems not actually possible since I have asked her once earlier and got no answer… If this is the situation, I would ask her to give me a new topic which actually comes from physics and for which one can actually work on any unknown issues.

But before I meet her, I would ask two questions to my confusion:

1. Does anyone know some real world models which indeed use restricted test functions like first order linear differential restrictions as described above?

2. Any advices for how should I talk to my advisor?

Any kind of help is welcome!