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A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

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Abstract

We consider a singular integral operator as a natural generalization to the biharmonic operator that arises in thin plate theory. The operator is built in the nonlocal calculus framework defined in (Math Models Methods Appl Sci 23(03):493–540, 2013) and connects with the recent theory of peridynamics. This framework enables us to consider non-smooth approximations to fourth-order elliptic boundary-value problems. For these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well-defined even for irregular domains. We demonstrate the existence and uniqueness of solutions to these nonlocal problems and demonstrate their L 2-strong convergence to functions in W 2,2 as the nonlocal interaction horizon goes to zero. For regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary-value problems. As a part of our proof we also establish that the nonlocal Laplacian of a smooth function is Lipschitz continuous.

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Authors and Affiliations

  1. Department of Mathematics, University of Nebraska-Lincoln, Lincoln, USA

    Petronela Radu & Daniel Toundykov

  2. Institute for Mathematical Sciences, The George Washington University, Washington, USA

    Jeremy Trageser

Authors
  1. Petronela Radu
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  2. Daniel Toundykov
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  3. Jeremy Trageser
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Corresponding author

Correspondence to Jeremy Trageser.

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Communicated by D. Kinderlehrer

The research of the first author was partially supported by the National Science Foundation under Grant DMS-0908435.

The research of second author was partially supported by the National Science Foundation under Grant DMS-1211232.

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Radu, P., Toundykov, D. & Trageser, J. A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue. Arch Rational Mech Anal 223, 845–880 (2017). https://doi.org/10.1007/s00205-016-1047-2

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  • DOI: https://doi.org/10.1007/s00205-016-1047-2

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