Abstract In this paper we initiate the study of second-order variational problems in L ∞ {L^{\infty}} , seeking to minimise the L ∞ {L^{\infty}} 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 s n × n ) {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})} , for the functional E ∞ ⁢ ( u , 𝒪 ) = ∥ H ⁢ ( D 2 ⁢ u ) ∥ L ∞ ⁢ ( 𝒪 ) , u ∈ W 2 , ∞ ⁢ ( Ω ) , 𝒪 ⊆ Ω , \mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{} the associated equation is the fully nonlinear third-order PDE A ∞ 2 ⁢ u := ( H X ⁢ ( D 2 ⁢ u ) ) ⊗ 3 : ( D 3 ⁢ u ) ⊗ 2 = 0 . \mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{} Special cases arise when H {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞ {\infty} -polylaplacian and the ∞ {\infty} -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.

中文翻译：

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二阶 L ∞ 变分问题和 ∞-polylaplacian

摘要 在本文中，我们开始研究 L ∞ {L^{\infty}} 中的二阶变分问题，寻求最小化 Hessian 函数的 L ∞ {L^{\infty}} 范数。我们还推导出并研究了作为欧拉-拉格朗日方程的类似物出现的相应偏微分方程。给定 H ∈ C 1 ⁢ ( ℝ sn × n ) {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})} ，对于泛函 E ∞ ⁢ ( u , 𝒪 ) = ∥ H ⁢ ( D 2 ⁢ u ) ∥ L ∞ ⁢ ( 𝒪 ) , u ∈ W 2 , ∞ ⁢ ( Ω ) , 𝒪 ⊆ Ω , \mathrm u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{} 相关方程是完全非线性的三阶偏微分方程 A ∞ 2 ⁢ u := ( HX ⁢ ( D 2 ⁢ u ) ) ⊗ 3 : ( D 3 ⁢ u ) ⊗ 2 = 0 。\mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \ mathrm{D}^{3}u)^{\otimes 2}=0。{} 当 H {\mathrm{H}} 是完整 Hessian 或 Laplacian 的欧几里得长度时，会出现特殊情况，分别导致 ∞ {\infty} -polylaplacian 和 ∞ {\infty} -bilaplacian。我们为 (1) 和 (2) 建立了几个结果，包括存在最小值、绝对最小值和“临界点”广义解，还证明了变分特征和唯一性。我们还构建了明确的广义解决方案并进行了数值实验。还证明了变分特征和唯一性。我们还构建了明确的广义解决方案并进行了数值实验。还证明了变分特征和唯一性。我们还构建了明确的广义解并进行了数值实验。