The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix section about the Banach space $L^p(\Omega)$. Could anybody come up with a cited reference for a careful treatment of this space?
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7$\begingroup$ Without other specification, it should denote the $L^p$ space on $\partial\Omega$ w.r.to the codimension 1 Hausdorff measure. $\endgroup$– Pietro MajerCommented Jan 6, 2016 at 7:51
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2$\begingroup$ I'm guessing $\Omega$ should be nice enough that surface area is defined. For example, $\Omega$ is a Lipschitz domain. Often PDE boundary conditions involve "normal vectors". $\endgroup$– Gerald EdgarCommented Jan 10, 2016 at 13:35
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2 Answers
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Check the book Theoretical Numerical Analysis by Kendall and Atkinson, and the book Functional Spaces for the Theory of Elliptic Partial Differential Equations by Demengel and Demengel.
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$\begingroup$ Would you give the name of the books? $\endgroup$– user14319Commented Jan 25, 2016 at 20:48
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You can find a lot of references about this, some books treat it like folklore since in most applications for PDEs it is the case. But, you can go as far as you want into how relaxed can be the boundary of the domain, $\partial \Omega$, in order to still have some nice theorems like Sobolev injections, trace operators, extension operators, Poincaré inequalities, etc.
If you want to know in some detail how the regularity of the boundary affects all the theorems I mentioned above and finally the existence and regularity of PDEs, the books from Demengel^2 and Trudinger and Gilbarg propose you a more serious approach. Not as deep as fractal domains but more than just a phrase saying you that it is a Lebesgue space with a Hausdorff measure.
If you are aiming to work with PDEs, maybe you will lose your time if you can live with the classical (almost folkloric) phrase : if you do it for a half space, then for a Lipschitz domain it is automatically done since you can glue together piece by piece with a partition of unity.
If you need more than this, you can read the above books and find a little bit of spiritual peace to advance in PDEs, but even for Lipschitz domains it could take you some time if you want more and more details (for example the Area formula).
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$\begingroup$ @Willie the typo is corrected. Thanks $\endgroup$– MarioCommented Sep 14, 2016 at 19:53