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Hello,

I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying

$ \| f \|_p < \infty, \|Df \|_q < \infty, $

where $p \ne q$ (as opposed to the alternate usage signifying that a different Sobolev exponent is imposed on normal versus tangential derivatives). In particular, I am interested in what kinds of trace and embedding results may have been proved about these function spaces. If anyone could suggest a good reference (or even another name by which such function spaces are known), I'd be very grateful. Thanks!

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    $\begingroup$ Use interpolation inequalities. $\endgroup$
    – Liviu Nicolaescu
    Commented Jan 8, 2013 at 19:44
  • $\begingroup$ Have you consulted, say, Adams' book on Sobolev spaces? Whatever you want to know is a straightforward consequences of the standard theorems on standard Sobolev spaces, using if needed the interpolation inequalities (i.e., Gagliardo-Nirenberg inequalities) cited by Liviu. $\endgroup$
    – Deane Yang
    Commented Jan 8, 2013 at 20:14

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You can find this in two of my following papers

MR3296206 Reviewed Han, Qi Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions. Monatsh. Math. 176 (2015), no. 1, 107–141. (Reviewer: Dimitri Mugnai) 35J66 (35B09 35B33 35J20 35J91 46E22 46E35)

http://www.sciencedirect.com/science/article/pii/S0007449715000913

and my email address is provided there. If you have any questions, please let me know and I am happy to discuss more with you.

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  • $\begingroup$ Thank you!! This is great, and your offer to answer questions is especially nice! $\endgroup$
    – Idempotent
    Commented Apr 9, 2016 at 18:42

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