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Mixed Biharmonic Problem with the Steklov-type and Neumann Boundary Conditions in Unbounded Domains

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Abstract

A biharmonic problem with mixed Steklov-type and Neumann conditions on the boundary in the exterior of a compact set is considered under the assumption that the generalized solutions of this problem have a bounded weighted Dirichlet integral. To solve this biharmonic problem, we use the variational principle and, depending on the value of the parameter included in the weighted integral, we obtain uniqueness (non-uniqueness) theorems, or the same as obtaining exact formulas for calculating the dimension of the solution space.

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Author information

Authors and Affiliations

  1. Universita di Parma, 43100, Parma, Italy

    Giovanni Migliaccio

  2. Federal Research Center ‘‘Computer Science and Control’’, Russian Academy of Sciences, 119333, Moscow, Russia

    Hovik A. Matevossian

  3. Moscow Aviation Institute (National Research University ‘‘MAI’’), 125993, Moscow, Russia

    Hovik A. Matevossian

Authors
  1. Giovanni Migliaccio
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  2. Hovik A. Matevossian
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Corresponding authors

Correspondence to Giovanni Migliaccio or Hovik A. Matevossian.

Additional information

(Submitted by A. M. Elizarov)

APPENDIX

APPENDIX

In [21] and [22], the authors derive a mathematical model used in mechanical engineering and to describe the radar process, as well as some applications in medicine. For solving these biharmonic problems with application, we need to solve boundary value problems for the Poisson equation using the scattering model. In order to select suitable solutions, we solve the Poisson equation with the corresponding boundary conditions, that is, some criterion function is minimized in the Sobolev norms.

Partial differential equations with Neumann-type boundary conditions represent the prototypical analytical model of many physical problems of interest in the field of structural mechanics. For example, the state of stress and strain of many structural elements, such as the load-bearing components of wind turbine blades, aircraft wings and civil buildings, can be obtained as solution of PDE problems derived via variational principles. This is the case, for instance, of the non-prismatic beamlike structures addressed in recent papers [23, 24]. Unfortunately, such PDE problems (which can always be solved numerically) admit closed-form analytical solutions only in a few cases, as in classical cases of the linear theory of prismatic beams and in the tapered and pre-twisted cases studied in [25]. However, apart from directly solving such problems, the corresponding PDEs often allow us to derive indirect yet important information on the state of stress and strain of the associated structural elements, as is also shown in [26] via paradigmatic examples of interest for engineering applications.

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Migliaccio, G., Matevossian, H.A. Mixed Biharmonic Problem with the Steklov-type and Neumann Boundary Conditions in Unbounded Domains. Lobachevskii J Math 43, 3222–3238 (2022). https://doi.org/10.1134/S1995080222140256

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  • DOI: https://doi.org/10.1134/S1995080222140256

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