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I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below). There, the authors consider domains of $R^n$ with regularity of class $W^2 L^{n-1,1}$(Sobolev-Lorentz Space). Unfortunately, despite of my efforts, I could not find any good reference about this kind of domains.

Then, I would like to ask if does anyone know a reference where the theory for manifolds of class $W^2 L^{n-1,1}$, $C^{1,1}$ or $W^{m,p}$ (ordinary Sobolev Space) is discussed in detail. In fact, I want to know, at least, how can be defined the mean curvature for these classes of manifolds.

The paper (really good one) is:

Ciachi and Maz'ya, Global Lipschitz Regularity For a Class of Quasilinear Elliptic Equations, CPDE, 36, 100-133,20

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  • $\begingroup$ Mean curvature is defined only for a submanifold of a Riemannian manifold. Do you want the ambient manifold to be a "Sobolev manifold" or the submanifold? Perhaps you could provide some more details or motivation for your question? $\endgroup$
    – Deane Yang
    Commented Nov 1, 2011 at 3:57
  • $\begingroup$ Hi Mr. Yang, Let me be more clear. I want to know some references when the ambient manifold is smooth (for instance $\mathbb{R}^n$) and the submanifolds are Sobolev regular. My motivation was the simplest case, where the submanifold is in fact a hypersurface of $\mathbb{R}^n$, the boundary of a domain(the 'classic' PDE overview). This kind of issue usually appear when working with regularity theory for PDE's. Cheers, Luís. $\endgroup$
    – deMiranda
    Commented Nov 1, 2011 at 20:32

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