Operational approach for biharmonic equations in \({\varvec{L}}^{{\varvec{p}}}\)-spaces


In this work, we study the existence, uniqueness and maximal \(L^p\)-regularity of the solution of different biharmonic problems. We rewrite these problems by a fourth-order operational equation and different boundary conditions, set in a cylindrical n-dimensional spatial region \(\Omega \) of \({\mathbb {R}}^n\). To this end, we give an explicit representation formula, using analytic semigroups, and invert explicitly a determinant operator in \(L^p\)-spaces thanks to \(\mathcal {E}_\infty \) functional calculus and operator sums theory.

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This research is supported by CIFRE Contract 2014/1307 with Qualiom Eco company. The author would like to thank the referee for the valuable and useful comments and corrections which help to improve this paper.

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Correspondence to Alexandre Thorel.

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Thorel, A. Operational approach for biharmonic equations in \({\varvec{L}}^{{\varvec{p}}}\)-spaces. J. Evol. Equ. 20, 631–657 (2020). https://doi.org/10.1007/s00028-019-00536-2

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Mathematics Subject Classification

  • 35B65
  • 35C15
  • 35J40
  • 35R20
  • 47A60
  • 47D06


  • Operational differential equations
  • Functional calculus
  • Analytic semigroups
  • Interpolation spaces
  • Maximal regularity