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I'm looking for some results about elliptic regularity with mixed boundary conditions. I know they exist with non mixed boundary conditions but where can I find some results for the mixed case? Thanks

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It's a fairly specialized area. I think most of the treatments of mixed boundary conditions would be in journal articles. As far as textbooks go, all I can find is these relatively brief discussions.

Carlo Miranda, "Partial differential equations of elliptic type", Springer 1970, pages 233–234 and 261.

Gilbarg and Trudinger, "Elliptic partial differential equations of second order", 3rd edition, Springer 1983, 1998, 2001, pages 47 and 215.

However, these just summarize some results and refer the reader to numerous journal articles.

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    $\begingroup$ thanks but I need something more specific, the idea is to find elliptic regularity estimation like the one for the following problem −Lapl(u) = f in Omega grad(u) n= g on Lambda so you have ||u||_H^2(Omega) < Cost || f ||_L^2(Omega) + || g ||_H^{1/2}(􀀀Lambda). Do you know some result? $\endgroup$
    – alison
    Oct 27, 2015 at 14:05
  • $\begingroup$ do you know some result for the mixed case of the same form of the previous inequality? $\endgroup$
    – alison
    Oct 27, 2015 at 14:07
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    $\begingroup$ The book of Lions and Magenes (vol 1) has what you need. $\endgroup$ Oct 27, 2015 at 15:18

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