On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

Abstract

The ∞‐Bilaplacian is a third‐order fully nonlinear PDE given by Δ∞2u≔()Δu3||normalD()Δu2=0. In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian Δp2u≔Δ||Δup−2Δu=0. We prove convergence of the numerical solution to the weak solution of Δp2u=0 and show that we are able to pass to the limit p → ∞. We perform various tests aimed at understanding the nature of solutions of Δ∞2u and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of D‐solutions.

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

“…In particular, some of the explicit solutions of the corresponding reduced ODEs are related to the results of numerical experimentation in Ref. 16. The complete classification is left as a future work.…”

“…In Ref. 16, equations of this type and the structure of their solutions were studied using appropriate numerical schemes. One of the goals of this paper is to construct new closed form solutions complementing these results.…”

Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

“…Diffuse derivatives can be seen as measure-theoretic disintegrations whose barycentres are the distributional derivatives (see [29]). For further results relevant to D-solutions and their applications, see [30]- [32], [34,10,35,13], [36]- [38].…”

An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

“…Herein we follow an approach based on recent advances in Calculus of Variations in the space L ∞ (see [22,23,24,25]) developed recently for functionals involving higher order derivatives. The field has been initiated in the 1960s by Gunnar Aronsson (see e.g.…”