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 auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained $L^\infty$ variational problem.
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 study the minimisation of a cost functional which measures the misfit on the boundary of a domain between a component of the solution to a certain parametric elliptic PDE system and a prediction of the values of this solution. We pose this problem as a PDE-constrained minimisation problem for a supremal cost functional in ${\mathrm{L}}^\infty$, where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in ${\mathrm{L}}^p$ as $p\to\infty$ and the generalised Kuhn-Tucker theory to derive the rel…
We prove the existence of vectorial Absolute Minimisers in the sense of Aronsson to the supremal functional $E_\infty(u,Ω') = \|\mathscr{L}(\cdot,u,D u)\|_{L^\infty(Ω')}$, $Ω'\Subset Ω$, applied to $W^{1,\infty}$ maps $u:Ω\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$ with given boundary values. The assumptions on $\mathscr{L}($ are minimal, improving earlier existence results previously established by Barron-Jensen-Wang and by the second author.
We show that for any regular bounded domain $Ω\subseteq \mathbb R^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial Ω$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.
Given the supremal functional $E_\infty(u,Ω')=ess\,\sup_{Ω'} H(\cdot,D u)$ defined on $W^{1,\infty}_{loc}(Ω,\mathbb{R}^N)$, $Ω' \Subset Ω\subseteq \mathbb{R}^n$, we identify a class of vectorial rank-one Absolute Minimisers by proving a statement slightly stronger than the next claim: vectorial solutions of the Hamilton-Jacobi equation $H(\cdot,D u)=c$ are rank-one Absolute Minimisers if they are $C^1$. Our minimality notion is a generalisation of the classical $L^\infty$ variational principle of Aronsson to the vector case and emerged in earlier work of the author. The assumptions are minimal…
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,…
Consider the supremal functional \[ \tag{1} \label{1} E_\infty(u,A) \,:=\, \|L(\cdot,u,D u)\|_{L^\infty(A)},\quad A\subseteq Ω, \] applied to $W^{1,\infty}$ maps $u:Ω\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$, $N\geq 1$. Under certain assumptions on $L$, we prove for any given boundary data the existence of a map which is: i) a vectorial Absolute Minimiser of \eqref{1} in the sense of Aronsson, ii) a generalised solution to the ODE system associated to \eqref{1} as the analogue of the Euler-Lagrange equations, iii) a limit of minimisers of the respective $L^p$ functionals as $p\rightarr…
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…
For $\mathrm{H} \in C^2(\mathbb{R}^{N \times n})$ and $u : Ω\subseteq \mathbb{R}^n \to \mathbb{R}^N$, consider the system \[ \label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \otimes \mathrm{H}\_P + \mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\mathrm{D} u): \mathrm{D}^2 u\, =\,0. \tag{1}\]We construct $\mathcal{D}$-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\mathcal{D}$-solutions are $W^{1,\infty}$-submersions and are obtained without any convexity hypotheses for $\mathrm{H}$, through a re…
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）
Let $n,N\in \mathbb{N}$ with $Ω\subseteq \mathbb{R}^n$ open. Given $H \in C^2(Ω\times \mathbb{R}^N\times \mathbb{R}^{Nn}),$ we consider the functional \[ \tag{1} \label{1} E_\infty (u,\mathcal{O})\, :=\, \underset{\mathcal{O}}{\mathrm{ess}\,\sup}\, H (\cdot,u,\mathrm{D} u) ,\ \ \ u\in W^{1,\infty}_\text{loc}(Ω,\mathbb{R}^N),\ \ \ \mathcal{O} \Subset Ω. \] The associated PDE system which plays the role of Euler-Lagrange equations in $L^\infty$ is \[ \label{2} \tag{2} \left\{ \begin{array}{r} H_{P}(\cdot, u, \mathrm{D}u)\, \mathrm{D} \big(H(\cdot, u, \mathrm{D} u)\big) \, = \, 0, \ \ \ H(\cdot, …
We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution $u:B_1\subset \mathbb{R}^n \to \mathbb{R}^m$ to the elliptic system \begin{equation*} \mbox{div} ((A + (B-A)χ_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous, uniformly elliptic matrices, we prove that if $\nabla u \in L^{\infty} (D)$ then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.
A periodic homogenization result of nonconvex integral functionals in the vectorial case with convex bounded constraints on gradients is proved. The class of integrands considered have singular behavior near the boundary of the convex set of the constraints. We apply the result to the case of periodic homogenization in hyperelasticity for bounded gradients of deformations.
Fluorescent Optical Tomography (FOT) is a new bio-medical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses non-ionising red and infrared light. Mathematically, FOT can be modelled as an inverse parameter identification problem, associated with a coupled elliptic system with Robin boundary conditions. Herein we utilise novel methods of Calculus of Variations in $L^\infty$ to lay the mathematical foundations of FOT which we pose as a PDE-constrained minimisation problem in $L^p$ and $L…
In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov re…
In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$, $Λ>0$ and $φ_i\in H^{1/2}(\partial D)$, we deal with \[ \min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+Λ\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=φ_i\;\mbox{on }\partial D\right\}}. \] We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ is made of a regular part, which is relatively open and loc…
We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},dμ)} \leq c \|μ\|_{L^{1,n-s}(\mathbb{R}^{n})}^{\frac{n-1}{n-s}}\|\mathbb{A}[D]u\|_{L^{1}(\mathbb{R}^{n},d\mathscr{L}^{n})}$$ hold for vector fields $u\in C^{\infty}(\mathbb{R}^{n};\mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $\mathbb{A}[D]$ on $\mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $\mathbb{A}[D]$ are necessary. Here, $\|μ\|_{L^{1,λ}}$ denotes the $(1,λ)$-Morrey norm of th…
In this paper we studythe asymptotics of singularly perturbed phase-transition functionals of the form \[ F_k(u)=\frac{1}{ε_k}\int_A f_k(x,u,ε_k\nabla u)\,dx\,, \] where $u \in [0,1]$ is a phase-field variable, $ε_k>0$ a singular-perturbation parameter, i.e., $ε_k \to 0$, as $k\to +\infty$, and the integrands $f_k$ are such that, for every $x$ and every $k$, $f_k(x,\cdot ,0)$ is a double well potential with zeros at 0 and 1. We prove that the functionals $F_k$ $Γ$-converge (up to subsequences) to a surface functional of the form \[ F_\infty(u)=\int_{S_u\cap A}f_\infty(x,ν_u)\,d\mathcal H^{n-1}…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of Distributions to PDEs and is not based on either integration by parts or on the maximum principle. Instead, our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the Young measures over a toric compactification of the space of jets. After developing some basic theory, as a first application we consider…
Let $H \in C^2(\mathbb{R}^{N \times n})$, $H\geq 0$. The PDE system \[ \label{1} A_\infty u \, :=\, \Big(H_P \otimes H_P + H [H_P]^\bot H_{PP} \Big)(Du) : D^2 u\, = \, 0 \tag{1} \] arises as the ``Euler-Lagrange PDE" of vectorial variational problems for the functional $E_{\infty}(u,Ω) = \| H(Du) \|_{L^\infty(Ω)}$ defined on maps $u : Ω\subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$. \eqref{1} first appeared in the author's recent work \cite{K3}. The scalar case though has a long history initiated by Aronsson in \cite{A1}. Herein we study the solutions of \eqref{1} with emphasis on the ca…
For a class of functionals having the $(p,q)$-growth, we establish an improved range of exponents $p$, $q$ for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the $L^{\infty}$ bound on the function rather than the $L^p$ estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak…
