广义高阶向量∞-特征值问题
Generalised higher order vectorial $\infty$-eigenvalue problems
AI总结：
本文针对任意整数k≥2的高阶向量∞-特征值问题，施加k阶类似固支和铰支的边界条件，通过L^p逼近方法证明特殊L^∞极小化子的存在性，确定特征值下界，将Clark和Katzourakis的二阶向量结果扩展到高阶情形。
AI中文摘要：
我们研究对于任意整数k≥2，在一类映射上最小化其k阶导数的函数的L^∞范数的问题，约束条件涉及该映射及其低阶导数的函数的L^∞范数。我们施加对应于经典“固支”和“铰支”情况的k阶类似物的边界条件。通过采用L^p逼近方法，我们确定了一个特殊的L^∞极小化子的存在性，该极小化子求解带有测度系数作为参数的散度型偏微分方程组，该方程组构成了所考虑的约束变分问题的Aronsson-Euler方程的对应物。此外，我们确定了该特征值的一个下界。本工作将Clark和Katzourakis的二阶向量结果（《广义二阶向量∞-特征值问题》，PRSE A，1-21，2024）扩展到了一般高阶情形。
英文摘要：
We study the problem of minimising the $L^\infty$ norm of a function of the $k$-th derivative over a class of maps, subject to a constraint involving the $L^\infty$ norm of a function of the map and its lower-order derivatives, for any integer $k\geq 2$. We impose boundary conditions corresponding to the $k$-th order analogues of the classical ``clamped'' and ``hinged'' cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimiser, which solves a divergence PDE system with measure coefficients as parameters. This system constitutes the counterpart of the Aronsson--Euler equations for the constrained variational problem under consideration. Furthermore, we establish a lower bound for the eigenvalue. The present work extends the second-order vectorial results of Clark and Katzourakis [Generalised second order vectorial $\infty$-eigenvalue problems, PRSE A, 1-21, 2024] to the general higher-order setting.
