We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of absolute minimisers and the concept of $\infty$-minimal maps introduced more recently by the second author. We prove that $C^1$ absolute minimisers characterise a divergence system with parameters probability measures and that $C^2$ $\infty$-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.
An introduction to the vectorial Calculus of Variations in $\textbf{L}^\infty$ AND Aronsson PDE systems
Department of Physics, College of Education for Pure Sciences, University of Kerbala（卡尔巴拉大学纯科学学院教育学院物理系）；Department of Mathematics and Statistics, University of Reading（雷丁大学数学与统计系）
We link the problem of estimating the lower Hausdorff dimension of PDE or Fourier constrained measures with Harnack's inequalities for the heat equation. Our approach provides new estimates in the case of Fourier constraints.
We study minimisation problems in $L^\infty$ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, jacobian and null Lagrangian constraints. Via the method of $L^p$ approximations as $p\to \infty$, we illustrate the existence of a special $L^\infty$ minimiser which solves a divergence PDE system involving certain…
We study a minimisation problem in $L^p$ and $L^\infty$ for certain cost functionals, where the class of admissible mappings is constrained by the Navier-Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all $p$, and also that $L^p$ minimisers converge to $L^\infty$ minimisers as $p\to\infty$. We further show that $L^p$ minimisers solve an Euler-Lagrange system. Finally, all special $L^\infty$ minimisers constructed via approximation by …
We propose a theory of non-differentiable solutions which applies to fully nonlinear PDE systems and extends the theory of viscosity solutions of Crandall-Ishii-Lions to the vectorial case. Our key ingredient is the discovery of a notion of extremum for maps which extends min-max and allows "nonlinear passage of derivatives" to test maps. This new PDE approach supports certain stability and convergence results, preserving some basic features of the scalar viscosity counterpart. In this first part of our two-part work we introduce and study the rudiments of this theory, leaving applications for…
Let $Ω\Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(Ω;\mathbb R^N)$ that satisfies \[ \big\| f(\mathrm D u) \big\|_{L^\infty(Ω)} \! = \inf \Big\{\big\| f(\mathrm D v) \big\|_{L^\infty(Ω)} \! : \ v \! \in W^{1,\infty}_0(Ω;\mathbb R^N), \, \| g(v) \|_{L^\infty(Ω)}\! =1\Big\}, \] under natural assumptions on $f,g$. This includes the $\infty$-eigenvalue problem as a special case. Herein we prove existence of a minimiser $u_\infty$ with extra properties, derived as t…
Characterisation of homogenisation for nonlocal diffusion by local topologies
TU Hamburg（汉堡工业大学）；School of Applied Mathematics and Informatics, University of Osijek（奥西耶克大学应用数学与信息学院）；Department of Mathematics, Faculty of Science, University of Zagreb（萨格勒布大学理学院数学系）；TU Bergakademie Freiberg（弗莱贝格矿业技术大学）
