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Analysis of a Mogi-type model describing surface deformations induced by a magma chamber embedded in an elastic half-space. (English. French summary) Zbl 1375.35535
The purpose of this article is to give a rigorous mathematical analysis of the formulas for the boundary displacement vector field of a spherical cavity. First, an integral representation formula is derived for the corresponding PDE with boundary conditions by using Betti’s formulas and the fundamental solution of the Lamé system. A solution to this problem is estimated by using the properties of homogeneous functions and their derivatives and spherical coordinates for distributions. In order to prove the uniqueness of the solution, the bounded inverse theorem, Ascoli-Arzéla theorem and Cauchy-Schwarz inequality are used. Finally, the main result, the asymptotic expansion of the solution, is proved by using the Taylor expansion for the Neumann function and the divergence theorem. Strangely enough, no Bessel functions turned up, despite the cylindrical symmetry of the problem.

MSC:
35Q74 PDEs in connection with mechanics of deformable solids
74B10 Linear elasticity with initial stresses
35C20 Asymptotic expansions of solutions to PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35J25 Boundary value problems for second-order elliptic equations
86A60 Geological problems
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