Applications of PDEs in functional spaces with a measure or a weight

In my presentation I want to discuss two problems to which applying methods of PDEs in functional spaces that depend on a certain kind of measure seems to be fruitful.
Describing a process of transfer of heat in a conductor that is made of parts of different dimensions is quite problematic if we rely only on standard methods of partial differential equations. This is due to the fact that we need somehow to deal with solutions supported on sets of different nature. It turns out that an application of the theory of differential equations in Sobolev spaces equipped with a certain kind of singular measure can give some interesting results. I plan to consider elliptic and parabolic problems in this setting.
In the second part, I will say a few words on the issue of the essential self-adjointness of a certain operator in weighted spaces. I want to discuss how the problem of determining essential self-adjointness is related to the regularity theory in Sobolev spaces with weights.

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