2018 # 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 …

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“…Putting all these together, it can be verified that $${\mathrm{normal\Pi}}_{\infty}^{2}false(a{x}^{r}+b{y}^{r}false)\sim {a}^{5}{x}^{5r-12}+{b}^{5}{y}^{5r-12}$$ from where we obtain, for $r=12\u22155$, a scaling invariant solution, also known as similarity solution , of Equation if and only if $(a,b)$ satisfies $${a}^{5}+{b}^{5}=0.$$ The only real solution is given by $b=-a$ and in this way we recover the solution $$u={x}^{12\u22155}-{y}^{12\u22155},$$ which was first constructed in Ref. . The same arguments are valid in the case of the general $\infty -$Polylaplacian in n independent variables.…”

confidence: 56%

“…Putting all these together, it can be verified that $${\mathrm{normal\Pi}}_{\infty}^{2}false(a{x}^{r}+b{y}^{r}false)\sim {a}^{5}{x}^{5r-12}+{b}^{5}{y}^{5r-12}$$ from where we obtain, for $r=12\u22155$, a scaling invariant solution, also known as similarity solution , of Equation if and only if $(a,b)$ satisfies $${a}^{5}+{b}^{5}=0.$$ The only real solution is given by $b=-a$ and in this way we recover the solution $$u={x}^{12\u22155}-{y}^{12\u22155},$$ which was first constructed in Ref. . The same arguments are valid in the case of the general $\infty -$Polylaplacian in n independent variables.…”

confidence: 56%

“…We also propose a conjecture for the symmetry structure of the general $\infty -$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 $\infty -$Polyharmonic functions and find potential minimizers for problems arising in the ${L}^{\infty}$ variational calculus.…”

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.…”

confidence: 99%

“…These functions are solutions of the ∞ ‐Bilaplacian which is the PDE $${\Delta}_{\infty}^{2}u\u2254{\left(\Delta u\right)}^{3}{\left(\mathrm{normalD}\left(\Delta u\right)\right)}^{2}=0,$$ 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 $$\mathcal{J}\left[u;\infty \right]\u2254{\left(\Delta u\right)}_{{\mathrm{normalL}}^{\infty}\left(\Omega \right)}.$$ …”

confidence: 99%

“…One possibility for a generalized solution concept to Equation is that of $\mathcal{D}$‐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.…”

confidence: 99%