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Self–adjoint Extensions for the Neumann Laplacian and Applications

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Abstract

A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so–called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self–adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains. The error estimates for proposed approximations of shape functionals are provided.

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Authors and Affiliations

  1. Institute of Mechanical Engineering Problems, V. O. Bolshoy pr. 61, 199178, St Petersburg, Russia

    S. A. Nazarov

  2. Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, 239, 54506, Vandoeuvre lés Nancy Cedex, France

    J. Sokołowski

  3. Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447, Warszawa, Poland

    J. Sokołowski

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  1. S. A. Nazarov
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Correspondence to S. A. Nazarov.

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Partially supported by INRIA in the framework of the grant 00–01 from Institut franco–russe A. M. Liapunov d’informatique et de mathématiques appliquées and by the grant 4 T11A 01524 of the State Committee for the Scientific Research of the Republic of Poland. The paper was prepared during a visit of S. A. Nazarov to the Institute Elie Cartan in Nancy

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Nazarov, S.A., Sokołowski, J. Self–adjoint Extensions for the Neumann Laplacian and Applications. Acta Math Sinica 22, 879–906 (2006). https://doi.org/10.1007/s10114-005-0652-z

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