Let me add some other references about elliptic regularity on singular manifolds:
- B.-W. Schulze and his co-authors have developed a comprehensive calculus for elliptic pseudo-differential operators (in particular differential operators) on singular spaces, including elliptic regularity and Fredholm properties:
1.1) For manifolds with conical singularities and manifolds with edges consider the following books:
B.-W. Schulze. Boundary Value Problems and Singular Pseudo-Differential Operators. J. Wiley, Chichester, 1998.
Ju.V. Egorov and B.-W. Schulze. Pseudo-Differential Operators, Singularities, Applications. Birkhäuser Verlag, Basel, 1997
1.2) For manifold with corners consider the following paper:
Schulze, Bert-Wolfgang. "The Mellin pseudo-differential calculus on manifolds with corners." Symposium “Analysis on Manifolds with Singularities”, Breitenbrunn 1990. Vieweg+ Teubner Verlag, 1992.
1.3) For manifolds with corner and edges the following is an upgraded version of the last paper:
B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publications of RIMS, Kyoto University {38}, 4 (2002), 735-802
1.4) For a more geometric treatment of elliptic operator on singular spaces consider:
Nazaikinskii, V. E., Savin, A. Y., Schulze, B. W., & Sternin, B. Y. (2005). Elliptic theory on singular manifolds. CRC Press.
In these papers/books they approach the problem with the goal of obtaining an algebra that contains elliptic operators and their parametrices (inverses module compact operators). If you are interested in a particular class of operators and need a concrete theory for that specific class you should also check the books of Mazya/Rossman/Kozlov:
Mazya, V. G., and J. Rossmann. Elliptic equations in polyhedral domains. No. 162. American Mathematical Soc., 2010. (this book might be particularly useful for the Laplacian on a cube)
Also for eigenvalues and spectral theory check this one:
Kozlov, Vladimir, Vladimir G. Mazí̂à, and Jürgen Rossmann. Spectral problems associated with corner singularities of solutions to elliptic equations. No. 85. American Mathematical Soc., 2001.