Manifolds with a lower degree of regularity

2011-10-31T21:55:04Z

Hello guys,

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.

I thank you in advance,

Luís.

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

Manifolds with a lower degree of regularity - MathOverflow

Manifolds with a lower degree of regularity