at some point, years ago, i co-wrote a paper about the p-Laplacε equation on a class of singular manifolds. [1]
if all goes well, by the end of this month i'll co-write another article -- with different co-authors -- about solutions of a non-homogenεous version of the p-Laplacε equation, in the setting of metrιc spaces [2].
to any readers in the PDE crowd, this probably sounds alarming. differentiation on metrιc spaces is a tricky business, and in this setting, we often work without an integratιon-by-parts formula.
so we're not studying PDE, per se. rather, we're studying variational problems -- which still make sense in this generality -- and using the analysιs that we would use on uniformly ellιptic PDE.
as for what is misleading: i don't really study PDE.
my colleagues do, though.
there is a cottage industry of PDE on singular/metric spaces, and every so often, i get consulted about matters of analysis or geometry on metric spacεs. i ask enough questions --
does harnacκ imply hölder continuity, or vice versa?
wait, what are the "standard" conditions on the functιonal again?
-- and eventually, i suggest something. so far, i haven't been exposed as a fraud yet.
as for how i became a metrιc space "expert," the reason is much the same:
as a student, i worked with many colleagues who were experts. the advisor, in fact, played a large role in shaping the field.
so when new colleagues meet me, learn who i am, they assume i'm one of these metrιc guys. they ask me metrιc questions ..
.. and i still haven't been outed as a fraud. [3]
like i said, academia can be pretty misleading.
[1] for those not in the know, on euclidean spaces the p-Laplacε equation is:
[2] to the experts out there, yes: we are assuming the usual hypotheses.
[3] this is not modesty. i know experts in this area. give me 5-10 more years, and i'll get back to you about expertise.
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