Second-order L ^∞ variational problems and the ∞-polylaplacian

Abstract

In this paper we initiate the study of 2nd 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 functionalthe associated equation is the fully nonlinear 3rd order PDESpecial cases arise when 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.

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

“…Putting all these together, it can be verified that from where we obtain, for , a scaling invariant solution, also known as similarity solution , of Equation if and only if satisfies The only real solution is given by and in this way we recover the solution which was first constructed in Ref. . The same arguments are valid in the case of the general Polylaplacian in n independent variables.…”

“…We also propose a conjecture for the symmetry structure of the general Polylaplacian in n dimensions. With this work, we aim to promote group theoretic ideas in the study of these strongly nonlinear problems and their exact solutions, i.e., the Polyharmonic functions and find potential minimizers for problems arising in the variational calculus.…”

Explicit $$\infty$$ ∞ -harmonic functions in high dimensions

“…[9]). Since then, the field has been developed enormously by N. Katzourakis in the series of papers ( [11][12][13][14][15][16][17][18][19]) and also in collaboration with the author, Abugirda, Croce, Manfredi, Moser, Parini, Pisante and Pryer ( [1,2,6,[20][21][22][23][24][25]). A standard difficulty of (1.1) is that it is nondivergence form equation and since in general smooth solutions do not exist, the definition of generalised solutions is an issue.…”

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

“…These functions are solutions of the ∞‐Bilaplacian which is the PDE derived in Ref. as the formal limit of the p ‐Bilaplacian as p → ∞. The ∞‐Bilaplacian is the prototypical example of a PDE from second‐order Calculus of Variations in L ∞ , arising as the analogue of the Euler–Lagrange equation associated with critical points of the supremal functional …”

“…One possibility for a generalized solution concept to Equation is that of ‐solutions . Roughly, this is a probabilistic approach where derivatives that do not exist classically are represented as limits of difference quotients into Young measures over a compactification of the space of derivatives.…”