# A note on regularity of solutions to degenerate elliptic by Felli V., Schneider S.

By Felli V., Schneider S.

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**Additional resources for A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type**

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23 page 23 October 12, 2015 10:9 24 9108 - Differential Geometry from a Singularity Theory Viewpoint 9789814590440 Differential Geometry from a Singularity Theory Viewpoint Let u = (u1 , . . , un ) denote the coordinates of a point in U with respect to the canonical basis of Rn . The tangent space Tp M of M at p = x(u) is an n-dimensional vector space generated by the linearly independent vectors of the partial derivatives of x at u. We take these vectors as a basis B(x) of Tp M , so B(x) = {xu1 (u), .

We have Nui (u) = −κ(u)xui (u), i = 1, . . , n, so for i, j = 1, . . , n, Nui uj (u) = −κuj (u)xui (u) − κ(u)xui uj (u). 3) The maps x and N are C ∞ -maps, therefore Nui uj (u) − Nuj ui (u) = 0 and xui uj (u) − xuj ui (u) = 0 for i, j = 1, . . , n. 3) that κuj (u)xui (u) − κui (u)xuj (u) = 0 for i, j = 1, . . , n. However, the vectors xu1 (u), . . , xun (u) are linearly independent, so κui (u) = 0 for i = 1, . . , n, that is, κ(u) is a constant function κ. We have two possibilities depending on whether κ is zero or not.

Furthermore, we have r βik = r βi δ k = =1 βi ν k , ν = νk , Gνui = νk , νui . =1 We have, from the definition of the coefficients of the ν-second fundamental form, s −hνi = νui , xu = Gνui , xu s αij xuj , xu = j=1 αij gj . = j=1 Thus, (αij ) = −(hνi )(g j )−1 = −(hji )ν . The expression for (π T ◦ Gν )ui follows from the fact that the vectors νi (u) are normal vectors. 3. The Lipschitz-Killing curvature of M at p = x(u) along the unit normal vector field ν is given by K ν (u) = det(hνij (u)) .