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Optimal elliptic Sobolev regularity near three-dimensional multi-material Neumann vertices. (English. Russian original) Zbl 1315.35077

Funct. Anal. Appl. 48, No. 3, 208-222 (2014); translation from Funkts. Anal Prilozh. 48, No. 3, 63-83 (2014).
In this paper, the authors study the optimal elliptic regularity (in \(W^{1,p}\)) 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 interest in such kind of problems comes from natural sciences and engineering. By applying techniques of localization, deformation and reflection, the Authors are able to prove that the gradient of any solution of the corresponding elliptic partial differential equation (in a neighborhood of the vertex) is \(p\)-integrable with \(p>3\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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