Geodesic
Optimal Elliptic Sobolev Regularity Near Three-Dimensional Multi-Material Neumann Vertices
Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 3, pp. 63-83

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We study the optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic div–grad operators in three dimensions at a multi-material vertex on the Neumann part of the boundary of a 3D polyhedral domain. The gradient of any solution of the corresponding elliptic partial differential equation (in a neighborhood of the vertex) is $p$-integrable with $p>3$.
Keywords: elliptic div–grad operator, piecewise linear 3D flattening, anisotropic ellipticity in three dimensions, transmission at material interfaces, mixed Dirichlet–Neumann boundary conditions, optimal Sobolev regularity. 
@article{FAA_2014_48_3_a5,
     author = {R. Haller-Dintelmann and W. H\"oppner and H.-Ch. Kaiser and J. Rehberg and G. M. Ziegler},
     title = {Optimal {Elliptic} {Sobolev} {Regularity} {Near} {Three-Dimensional} {Multi-Material} {Neumann} {Vertices}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {63--83},
     publisher = {mathdoc},
     volume = {48},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2014_48_3_a5/}
}
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R. Haller-Dintelmann; W. Höppner; H.-Ch. Kaiser; J. Rehberg; G. M. Ziegler. Optimal Elliptic Sobolev Regularity Near Three-Dimensional Multi-Material Neumann Vertices. Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 3, pp. 63-83. http://geodesic.mathdoc.fr/item/FAA_2014_48_3_a5/