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Regularization of quasilinear stationary problems related to the problem of large deformations of the membrane. (English, Russian) Zbl 1159.35435

Mosc. Univ. Comput. Math. Cybern. 2006, No. 1, 1-9 (2006); translation from Vestn. Mosk. Univ., Ser. XV 2006, No. 1, 5-12 (2006).
Summary: The work deals with a certain modification of the problem of equilibrium of an elastic membrane in the case of large deformations based on the fact that constraints of the type \(|\nabla v|\leq k\) are necessary for actual physical membranes. Therefore, when choosing a mathematical model, we can use this constraint (with a fairly large \(k\)) for replacing the function \(g(\xi)\equiv [1+\xi]^{1/2}\) for \(\xi>k\) by a more plausible function in order to simplify the correction analysis and the process of calculation of the solution of a new problem. In this case, the analysis in a Hilbert space of the form \(H\equiv H^1(\overline{\Omega}; \Gamma_0)\) becomes possible (\(H\) consists of elements from a Sobolev space \(W_2^1(\Omega)\) with zero traces on the interval \(\Gamma_0\) of the boundary of the domain). It is essential that the related problems are also analyzed in strengthened Sobolev spaces \(G\equiv G^{1,1}(\overline{\Omega};S)\) for problems on a large deformation of the membrane supported by a one-dimensional framework \(S\) (system of stringers). Special attention is given to the analysis of problems of this kind upon the increase of stiffness of stringers; if none of the stringers forms zero angles with other arcs contained in \(S\cup\Gamma_0\), then asymptotically optimal estimates of excitation are obtained. The employed reduction to the problem with a harmonically structurized operator is of a very considerable importance for constructing efficient projection-net methods and the corresponding iteration algorithms.

MSC:

35Q72 Other PDE from mechanics (MSC2000)
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
74K15 Membranes
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