2018
DOI: 10.1515/acv-2016-0052
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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 … Show more

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Cited by 23 publications
(37 citation statements)
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“…Putting all these together, it can be verified that normalΠ2false(axr+byrfalse)a5x5r12+b5y5r12from where we obtain, for r=125, a scaling invariant solution, also known as similarity solution , of Equation if and only if (a,b) satisfies a5+b5=0.The only real solution is given by b=a and in this way we recover the solution u=x125y125,which was first constructed in Ref. . The same arguments are valid in the case of the general Polylaplacian in n independent variables.…”
Section: Symmetry Reductions and Invariant Solutionsmentioning
confidence: 56%
See 1 more Smart Citation
“…Putting all these together, it can be verified that normalΠ2false(axr+byrfalse)a5x5r12+b5y5r12from where we obtain, for r=125, a scaling invariant solution, also known as similarity solution , of Equation if and only if (a,b) satisfies a5+b5=0.The only real solution is given by b=a and in this way we recover the solution u=x125y125,which was first constructed in Ref. . The same arguments are valid in the case of the general Polylaplacian in n independent variables.…”
Section: Symmetry Reductions and Invariant Solutionsmentioning
confidence: 56%
“…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 L variational calculus.…”
Section: Introductionmentioning
confidence: 99%
“…[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.…”
Section: Introductionmentioning
confidence: 99%
“…These functions are solutions of the ‐Bilaplacian which is the PDE Δ2u()Δu3||normalD()Δu2=0, 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 Ju‖‖ΔunormalLΩ. …”
Section: Introduction and The ∞‐Bilaplacianmentioning
confidence: 99%
“…One possibility for a generalized solution concept to Equation is that of D‐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.…”
Section: Introduction and The ∞‐Bilaplacianmentioning
confidence: 99%