Second-Order L ∞ Variational Problems and the ∞-Polylaplacian

Nikos Katzourakis, Tristan Pryer

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In this paper we initiate the study of second-order variational problems in L , seeking to minimise the L norm of a function of the hessian.We also derive and study the respective PDE arising as the analogue of the Euler Lagrange equation. Given H ∈ C 1(R n×n s ), for the functional E (u, O) = |H(D 2u)|L (O) , u ∈ W 2,∞(Ω), O ⊆ Ω, (1) the associated equation is the fully nonlinear third-order PDE A 2 u := (H X(D 2u)) ⊗3 : (D 3u) ⊗2 = 0. (2) Special cases arisewhen H is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞-polylaplacian and the ∞-bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

Original languageEnglish
Pages (from-to)115-140
Number of pages26
JournalAdvances in Calculus of Variations
Issue number2
Early online date27 Jan 2018
Publication statusPublished - 1 Apr 2020

Bibliographical note

Publisher Copyright:
© 2020 Walter de Gruyter GmbH, Berlin/Boston.


  • math.AP
  • math.NA
  • generalised solutions,fully nonlinear equations
  • Baire category method
  • convex integration
  • young measures

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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